Large Amplitude Quasi-Periodic Traveling Waves in Two Dimensional Forced Rotating Fluids

TL;DR

This paper proves the existence of large quasi-periodic traveling waves in two-dimensional rotating fluids under external forces.

Contribution

It presents the first construction of quasi-periodic solutions for a quasilinear PDE in higher dimensions with a highly degenerate dispersion relation.

Findings

Quasi-periodic traveling wave solutions exist for the β-plane equation with large amplitude.

The proof uses a nonlinear Nash-Moser scheme to handle small divisors and degeneracy.

The method preserves traveling-wave structure and uses normal form techniques for sublinear dispersion.

Abstract

We establish the existence of quasi-periodic traveling wave solutions for the \documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document}β-plane equation on \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}^2$$\end{document}T2 with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.

Full text

Introduction

Incompressible rotating fluids in three dimensions are described by the Euler-Coriolis equations, which, on the three dimensional torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^3 = ({\mathbb {R}}/2\pi {\mathbb {Z}})^3$$\end{document} , read as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\partial _{t} {{\textbf {u}}}+ ({{\textbf {u}}}\cdot \nabla ) {{\textbf {u}}}+ {\mathfrak {f}}\, \overrightarrow{e_3} \wedge {{\textbf {u}}}+ \nabla P = \textbf{F}, \\&\quad \nabla \cdot {{\textbf {u}}}=0, \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {u}}}(t, x): {\mathbb {R}}\times {\mathbb {T}}^3 \rightarrow {\mathbb {R}}^3$$\end{document} is the divergence-free velocity field, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(t, x): {\mathbb {R}}\times {\mathbb {T}}^3 \rightarrow {\mathbb {R}}$$\end{document} is the scalar pressure, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {f}}={\mathfrak {f}}(x_2)$$\end{document} is a scalar function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow{e_3}=(0,0,1)^\top $$\end{document} is the rotation vector, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{F}(t,x):{\mathbb {R}}\times {\mathbb {T}}^3 \rightarrow {\mathbb {R}}^3$$\end{document} is an external force. The term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {f}}\,\overrightarrow{e_3}\wedge {{\textbf {u}}}$$\end{document} is usually referred as the Coriolis force. It is customary in geophysical fluid dynamics to simplify the model introducing some approximations. First, noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overrightarrow{e_3}\wedge {{\textbf {u}}}= (-u_2,u_1,0)^\top := ({{\textbf {u}}}_h^\perp , 0)^\top $$\end{document} , according to the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {u}}}=(u_1,u_2,u_3)=: ({{\textbf {u}}}_h,u_3)$$\end{document} , we assume a trivial dependence with respect to the third direction, namely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{x_3} {{\textbf {u}}}=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{x_3}\textbf{F}= 0$$\end{document} . As part of the standard elliptic constraint \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \cdot \big (({{\textbf {u}}}\cdot \nabla ){{\textbf {u}}}+{\mathfrak {f}}\, \overrightarrow{e_3} \wedge {{\textbf {u}}}- \textbf{F}\big ) + \Delta P =0$$\end{document} on the pressure, we also assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{x_3} P =0$$\end{document} . This way, setting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla =(\partial _{x_1},\partial _{x_2},\partial _{x_3})^\top =: (\nabla _{h},\partial _{x_3})^\top , \quad \textbf{F}= ({\mathtt F}_{1},{\mathtt F}_{2},{\mathtt F}_{3})^\top =: (\textbf{F}_{h},{\mathtt F}_{3})^\top , \end{aligned}$$\end{document}

we are led to the reduced two dimensional problem

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\partial _{t} {{\textbf {u}}}_h + ({{\textbf {u}}}_h \cdot \nabla _h) {{\textbf {u}}}_h + {\mathfrak {f}}\, {{\textbf {u}}}_h^\perp + \nabla _{h} P = \textbf{F}_{h} , \\&\quad \nabla _{h} \cdot {{\textbf {u}}}_h=0. \end{aligned} \end{aligned}$$\end{document}

A solution to the original three dimensional problem (1.1) is then recovered by solving the linear transport equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial _{t} u_3 + {{\textbf {u}}}_h \cdot \nabla _{h} u_3 = {\mathtt F}_{3}$$\end{document} .

Next, we introduce the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane approximation by choosing

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathfrak {f}}(x_2)=\beta x_2, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in {\mathbb {R}}\setminus \{0\}$$\end{document} is a non-trivial constant. Note that we choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {f}}$$\end{document} to be a linear function, and the vertical dynamics are considered trivial, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{x_3}{{\textbf {u}}}= 0$$\end{document} as above. For further details, we refer to works such as McWilliams [40], Pedlosky [42], Chemin, Desjardins, Gallagher and Grenier [15], Gallagher and Saint-Raymond [29], and Pusateri and Widmayer [43]. The constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in {\mathbb {R}}\setminus \{0\}$$\end{document} represents the speed of rotation of the frame system around the rotation vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overrightarrow{e_3}$$\end{document} . Throughout this work, we treat \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} as a fixed constant.

Introducing the scalar vorticity (i.e. the third component of the vorticity field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \wedge {{\textbf {u}}}$$\end{document} )

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v:= \textrm{curl} \, {{\textbf {u}}}_h := \partial _{x_1} u_2 - \partial _{x_2} u_1, \end{aligned}$$\end{document}

and applying the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{curl}$$\end{document} operator to (1.2), yields the scalar equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t v + {{\textbf {u}}}_h \cdot \nabla _{h} v + \beta \, {\mathtt L}v = \textrm{curl}\, \textbf{F}_{h} ,\quad \textbf{u}_h=\nabla _{h}^\perp (-\Delta )^{-1} v , \end{aligned}$$\end{document}

which is the so-called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane equation, where we have, under the incompressibility condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{h}\cdot {{\textbf {u}}}_h = 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{curl}( {\mathfrak {f}}\,{{\textbf {u}}}_h^\perp )= \partial _{x_1}(\beta x_2 u_1) - \partial _{x_2}(-\beta x_2 u_2) = \beta u_2 = \beta \, {\mathtt L}v, \end{aligned}$$\end{document}

and the operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathtt L}:=- (-\Delta )^{-1} \partial _{x_1}}, \end{aligned}$$\end{document}

namely the second component of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{h}^\perp (-\Delta )^{-1}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{h}^\perp = (\partial _{x_2},-\partial _{x_1})^\top $$\end{document} , acts on any function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x):{\mathbb {T}}^2\rightarrow {\mathbb {R}}$$\end{document} with zero average as the Fourier multiplier

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt L}h(x):= \sum _{\xi \in {\mathbb {Z}}^2\setminus \{0\}} \textrm{i}\,{\mathtt L}(\xi ) {{\widehat{h}}}(\xi ) e^{\textrm{i}\,\xi \cdot x}, \quad { {\mathtt L}(\xi ):=- \frac{\xi _{1}}{|\xi |^2} }\quad \forall \,\xi =(\xi _{1},\xi _{2})^\top \in {\mathbb {Z}}^2\setminus \{0\}.\nonumber \\ \end{aligned}$$\end{document}

Remark 1.1

The sign in front of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} in equation (1.3) may be different compared to previous works in literature (for instance, see [21, 41, 43]). This discrepancy comes from how the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt L}$$\end{document} in (1.4)-(1.5) is defined, where a change of sign in front of the Laplacian appears depending on the sign convention for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{h}^\perp $$\end{document} .

Approximating models for rotating fluids in two and three dimensions are commonly used in oceanography and geophysical fluid dynamics. An interesting regime that is frequently studied is when the fluid is rapidly rotating, namely when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\beta | \gg 1$$\end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane approximation, even in the viscous case. In a series of works, Babin, Mahalov and Nicolaenko [2–4] proved regularity and integrability properties of solutions in 3D resonant tori. Dalibard and Saint-Raymond [19] devoted their analysis of the long-time behavior of solutions The singular limit in the vanishing Rossby number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Ro} \sim \beta ^{-1}$$\end{document} was investigated by Charve [14] and Dutrifoy [20]. The long-time behavior of solutions with high-speed rotation was studied by Koh, Lee and Takada [38] and Takada [46] by means of Strichartz estimates, by Dalibard [18] under the effect of random forcing and by Angullo-Castillo and Ferreira [1] in Besov spaces. The relation between the speed of rotation and the lifespan of solutions was investigated by Ghoul, Ibrahim, Lin and Titi [30] for the three dimensional primitive equations.

Independently of the speed of rotation, the global well-posedness of solutions was studied for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane equation by Elgindi and Widmayer [21] and Pusateri and Widmayer [43]. The global well-posedness for the Euler-Coriolis equations was studied by Guo, Huang, Pausader and Widmayer [32] and Guo, Pausader and Widmayer [33] with axisymmetric initial data, and, very recently, by Ren and Tian [44] with general non-axisymmetric initial data. For the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane equation, we mention also the works of Wei, Zhang and Zhu [48] on the linear inviscid damping around shear flows, and of Wang, Zhang and Zhu [47] on the existence of non-shear solutions close to the Couette flow depending on the size of the rotational speed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} .

Main result

Led by the physical model discussed above, in this paper we focus our attention on the effect of a large forcing term of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{curl}\, \textbf{F}_{h} (t,x) = \lambda ^{\alpha } f(\lambda \,\omega t,x) \end{aligned}$$\end{document}

where the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} is large, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} is a positive exponent and, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in {\mathbb {N}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \geqslant 2$$\end{document} , the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi },x)$$\end{document} is quasi-periodic in time with frequency vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \in {\mathbb {R}}^\nu \setminus \{0\}$$\end{document} , evaluated at the linear flow \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\mapsto f({\varphi },x)|_{{\varphi }= \lambda \, \omega t} $$\end{document} . We shall assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in {{\mathcal {C}}}^\infty ({\mathbb {T}}^\nu \times {\mathbb {T}}^2)$$\end{document} , $[eqn]$ , has zero average in x, that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {T}}^2} f({\varphi }, x)\, \,\textrm{d}{x} = 0, \quad \forall {\varphi }\in {\mathbb {T}}^\nu . \end{aligned}$$\end{document}

The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} will be fixed later in order to construct solutions of large amplitude.

Therefore, with a minor change in the notation, from now on we work with the following equation, on the two dimensional torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{t} v+ \beta \, {\mathtt L}v + u \cdot \nabla v = \lambda ^{\alpha } f(\lambda \,\omega t,x) , \quad t\in {\mathbb {R}}, \ x \in {\mathbb {T}}^2, \end{aligned}$$\end{document}

where the two dimensional velocity field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:{\mathbb {R}}\times {\mathbb {T}}^2 \rightarrow {\mathbb {R}}^2$$\end{document} is determined from the scalar vorticity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v: {\mathbb {R}}\times {\mathbb {T}}^2 \rightarrow {\mathbb {R}}$$\end{document} by the classical Biot-Savart law

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u(t,x):= {\mathfrak {B}}[v (t,x)]:= \nabla ^\perp (-\Delta )^{-1} v(t,x) = \sum _{\xi \in {\mathbb {Z}}^2\setminus \{0\}} {\frac{\textrm{i}(\xi _{2},-\xi _{1})^\top }{|\xi |^2} }{{\widehat{v}}}(t,\xi ) e^{\textrm{i}\, \xi \cdot x}.\nonumber \\ \end{aligned}$$\end{document}

With equation (1.7), we aim to study how the presence of a forcing term, of substantial magnitude and highly oscillating in time, affects the dynamics in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane approximation without relying on a perturbative regime. Additionally, we seek to determine whether the quasi-periodic structure of the external force produces a time quasi-periodic solution of (1.7), with substantial size and rapid oscillations, specifically with a frequency vector of oscillation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \in {\mathbb {R}}^\nu {\setminus }\{0\}$$\end{document} of significant magnitude. The interaction between the size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} of the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega $$\end{document} and the size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^\alpha $$\end{document} of the forcing term will be discussed later.

In particular, we search for quasi-periodic traveling wave solutions, according to the following definition.

Definition 1.2

(Quasi-periodic traveling wave). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in {\mathbb {N}}$$\end{document} . We say that a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v:{\mathbb {R}}\times {\mathbb {T}}^2\rightarrow {\mathbb {R}}$$\end{document} is a quasi-periodic traveling wave with irrational frequency vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega = (\omega _{1},...,\omega _{\nu })\in {\mathbb {R}}^{\nu }{\setminus }\{0\}$$\end{document} , that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \cdot \ell \ne 0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu \setminus \{0\}$$\end{document} , and “wave vectors” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\jmath }}_{1},...,{\overline{\jmath }}_{\nu }\in {\mathbb {Z}}^2$$\end{document} if there exists a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{v}:{\mathbb {T}}^\nu \rightarrow {\mathbb {R}}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v(t,x) = \breve{v} (\omega _{1}t- \overline{\jmath }_1\cdot x,...,\omega _{\nu }t-\overline{\jmath }_{\nu }\cdot x) = \breve{v}(\omega t- \pi ( x)) , \quad \forall (t,x) \in {\mathbb {R}}\times {\mathbb {T}}^2, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :{\mathbb {R}}^2 \rightarrow {\mathbb {R}}^\nu $$\end{document} is the linear map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto \pi ( x):= (\overline{\jmath }_{1}\cdot x,...,\overline{\jmath }_{\nu }\cdot x)$$\end{document} . We also denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top $$\end{document} the transpose of the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} .

We fix once and for all the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\jmath }}_{1},...,{\overline{\jmath }}_{\nu }\in {\mathbb {Z}}^2$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dim}\,\textrm{span}\{\overline{\jmath }_{1},...,\overline{\jmath }_{\nu }\} = 2$$\end{document} , in order to avoid trivial cases. Therefore, assuming that the forcing term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\lambda \,\omega t, x)$$\end{document} is a quasi-periodic traveling function according to Definition 1.2, for most values of the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathbb {R}}^\nu \setminus \{0\}$$\end{document} , we search for quasi-periodic traveling wave solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v({\varphi },x)|_{{\varphi }= \lambda \,\omega t}$$\end{document} with large frequency vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \in {\mathbb {R}}^{\nu }$$\end{document} (with a slight abuse of notation between v(t, x) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\lambda \,\omega t,x)$$\end{document} ) of the equations (1.7)-(1.8), which we rewrite as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \,\omega \cdot \partial _{{\varphi }} v + \beta \,{\mathtt L}v + u \cdot \nabla v = \lambda ^{\alpha } f(\lambda \, \omega t,x) , \quad u = {\mathfrak {B}}[v] = \nabla ^\perp (-\Delta )^{-1} v. \end{aligned}$$\end{document}

We look for solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \in H^s_0({\mathbb {T}}^\nu \times {\mathbb {T}}^2)$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \gg 1$$\end{document} large enough, where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} H^s({\mathbb {T}}^\nu \times {\mathbb {T}}^2)&:= \Big \{ u({\varphi }, x) = \sum _{(\ell , j) \in {\mathbb {Z}}^\nu \times {\mathbb {Z}}^2 } {\widehat{u}}(\ell , j) e^{\textrm{i}(\ell \cdot {\varphi }+ j \cdot x)}: \Vert u \Vert _s^2 \\&:= \sum _{(\ell , j) \in {\mathbb {Z}}^\nu \times {\mathbb {Z}}^2} \langle \ell , j \rangle ^{2 s} |{\widehat{u}}(\ell , j)|^2 < + \infty \Big \}, \\ H^s_0({\mathbb {T}}^\nu \times {\mathbb {T}}^2)&:= \Big \{ u \in H^s({\mathbb {T}}^\nu \times {\mathbb {T}}^2): \int _{{\mathbb {T}}^2} u({\varphi }, x) \, d x = 0 \Big \}, \\ \langle \ell , j \rangle&:= \max \{ 1, |\ell |, |j| \}. \end{aligned} \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^\nu $$\end{document} be the reference compact set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega := \{\omega \in {\mathbb {R}}^\nu : \, 1 \leqslant |\omega | \leqslant 2 \}. \end{aligned}$$\end{document}

The main theorem of this paper is the following.

Theorem 1.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in {\mathbb {R}}\setminus \{0\}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha \in (1,2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \in {\mathbb {N}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \geqslant 2$$\end{document} , be fixed. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in {{\mathcal {C}}}^\infty ({\mathbb {T}}^\nu \times {\mathbb {T}}^2, {\mathbb {R}})$$\end{document} be a quasi-periodic traveling wave (see Definition 1.2) satisfying (1.6). There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{s}} = {\overline{s}}(\nu , \alpha ) > 0$$\end{document} such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant {\overline{s}}(\nu , \alpha )$$\end{document} , there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _0 = \lambda _0(f, s, \nu , \alpha , \beta ) \gg 1$$\end{document} large enough and constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1, C_2>0$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_i = C_i(f, s, \nu , \alpha , \beta ) $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} , such that, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \geqslant \lambda _0$$\end{document} , the following holds. There exists a Borel set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\lambda \subset \Omega $$\end{document} of asymptotically full Lebesgue measure as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow + \infty $$\end{document} , that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{\lambda \rightarrow + \infty } |\Omega {\setminus } \Omega _\lambda | = 0$$\end{document} , such that, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\lambda $$\end{document} , there exists a quasi-periodic traveling wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\lambda (\,\cdot \,; \omega ) \in H^s_0({\mathbb {T}}^\nu \times {\mathbb {T}}^2)$$\end{document} that solves the equation (1.9). Moreover

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1 \lambda ^{\alpha -1} \leqslant \inf _{\omega \in \Omega _\lambda } \Vert v_\lambda (\,\cdot ; \omega ) \Vert _s \leqslant \sup _{\omega \in \Omega _\lambda } \Vert v_\lambda (\,\cdot ; \omega ) \Vert _s \leqslant C_2 \lambda ^{\alpha -1+{\mathtt c}}, \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}\in (0,{\mathtt c}_{0})$$\end{document} arbitrarily small, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}_{0}= {\mathtt c}_{0}(\alpha ,\nu )$$\end{document} . Finally, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi },x)$$\end{document} that is even in the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} , then the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\lambda }(\omega t,x)$$\end{document} is linearly stable.

With Theorem 1.3, we prove that the presence of a large external forcing term that is quasi-periodic traveling and with large frequency of oscillations induces solutions to the forced \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -plane equation (1.9) of large amplitude with the same properties for any value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} sufficiently large. Let us make some remarks on the result.

1. The role of the quasi-periodic traveling structure. When the free dynamics of a system (that is, without an external force) is invariant by translations in space, traveling wave functions, either periodic or quasi-periodic, form a natural class where to search a solution to the evolution problem that preserves this symmetry. In the context of the model considered here, it is actually an important structure that we need to impose in order to avoid potential degeneracies and multiplicity of the eigenvalues, already at the linear level. More specifically, the linear dynamics of the equation (1.9) is governed by the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}$$\end{document} , whose spectrum when acting on functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s({\mathbb {T}}^\nu \times {\mathbb {T}}^2)$$\end{document} is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{spec} \big ( \lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}\big ) = \big \{ \textrm{i}(\lambda \,\omega \cdot \ell +\beta \,{\mathtt L}(j) ) : \, \ell \in {\mathbb {Z}}^\nu , \ j =(j_1,j_2)^\top \in {\mathbb {Z}}^2 \big \}. \end{aligned}$$\end{document}

Assuming non-resonance conditions as the parameter varies (that shall be imposed later), the degeneracy is due to the fact the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \textrm{spec} \big ( \lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}\big )$$\end{document} has infinite multiplicity, since it corresponds to any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,j)=(0,(0,j_2)^\top )$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_2\in {\mathbb {Z}}$$\end{document} , recalling (1.5). When acting on quasi-periodic traveling waves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v({\varphi },x)=\breve{v}({\varphi }-\pi (x))$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)\in {\mathbb {T}}^\nu \times {\mathbb {T}}^2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }-\pi (x)\in {\mathbb {T}}^\nu $$\end{document} (see also Definition 2.3), it is worth noting that the average in the angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }\in {\mathbb {T}}^\nu $$\end{document} coincides with the average taken over both coordinates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} . This follows from the simple computation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {T}}^\nu } v({\varphi },x) \,\textrm{d}{{\varphi }}= \int _{{\mathbb {T}}^\nu } \breve{v}({\varphi }-\pi (x)) \,\textrm{d}{{\varphi }}= \int _{{\mathbb {T}}^\nu } \breve{v}({\varphi }) \,\textrm{d}{{\varphi }}= \text {constant w.r.t.}\,\, x. \end{aligned}$$\end{document}

At the Fourier level, this implies that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell =0$$\end{document} , then necessarily \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=0$$\end{document} as well. The previous degeneracy is avoided when the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}$$\end{document} acts on quasi-periodic traveling waves.

2. The role of the large frequency and of the large forcing term. The size of the forcing term is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\alpha }$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} . We consider the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} for the following reasons. At a purely heuristic level, one can expect the size of a (quasi-)periodic solution to be given by:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {size of the solution} = \frac{\text {size of the force}}{\text {size of the frequency}} = \frac{O(\lambda ^{\alpha })}{O(\lambda )} = O(\lambda ^{\alpha -1}) . \end{aligned}$$\end{document}

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} , the expected solution will be of small size with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1$$\end{document} , it will not be of small size, but uniformly bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} . The regime of interest is when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} . However, searching for solutions of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{\alpha -1})$$\end{document} implies that the quadratic nonlinearity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \cdot \nabla v $$\end{document} is of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{2\alpha -2})$$\end{document} . To apply perturbative arguments in our scheme, we need the size of the nonlinear terms to be smaller than the size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{\alpha })$$\end{document} of the external forcing term. This condition implies the restriction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <2$$\end{document} , which we apply. It is worth noting that we do not know if solutions in the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \geqslant 2$$\end{document} might exist or not: handling such cases would likely require different strategies and techniques. Moreover, since the construction of the solution deals with small divisors and relative non-resonances conditions, the actual estimates for the solutions do not follow exactly the heuristic in (1.13), but we have to slightly worsen the rate for the upper bound, as stated in (1.12).

We also want to emphasize that the threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _0(f,s,\nu ,\beta )\gg 1$$\end{document} in Theorem (1.3) needs to be strictly larger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\beta |>0$$\end{document} . This condition is crucial to obtain solutions of effective large size, satisfying the lower bound in (1.12). The solutions will exist for any fixed value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in {\mathbb {R}}\setminus \{0\}$$\end{document} , regardless of its size. However, it is essential to note that these solutions will exhibit large amplitude only if the fast external oscillations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega $$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , are stronger than the internal speed of rotation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \ne 0$$\end{document} .

3. Reversible solutions and linear stability. It is possible to show that the free dynamics, namely equation (1.7) without the external forcing term, is reversible, which is a reflection of the well-known property of the Euler-Coriolis equations. If we assume parity conditions on the external forcing, in particular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi },x)$$\end{document} being even in the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} , then the equation (1.9) is still reversible and we can adapt our scheme to produce solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\lambda }$$\end{document} in Theorem 1.3 that are odd in the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} . Moreover, we can prove under this extra symmetry that these quasi-periodic traveling waves solutions are linearly stable, meaning that the Floquet exponent of the linearized operator at each solution, with the action of the latter restricted to quasi-periodic traveling waves, are purely imaginary. However, the symmetry of the reversible structure is not necessary for the construction of the solutions in Theorem (1.3) itself, but rather only for the analysis of their stability.

Literature on quasi-periodic waves in fluids

In the last years, there has been a discrete surge of works proving the existence of time quasi-periodic waves for PDEs arising in fluid dynamics. Most of these results in literature are proved by means of KAM for PDEs techniques, to deal with the presence of small divisors issues and consequent losses of regularity. For the two dimensional water waves equations, we mention Berti and Montalto [11], Baldi, Berti, Haus and Montalto [5] for time quasi-periodic standing waves and Berti, Franzoi and Maspero [8, 9], Feola and Giuliani [23] for time quasi-periodic traveling wave solutions. Recently, the existence of time quasi-periodic solutions was proved for the contour dynamics of vortex patches in active scalar equations. We mention Berti, Hassainia and Masmoudi [10] for vortex patches of the Euler equations close to Kirchhoff ellipses, Hmidi and Roulley [37] for the quasi-geostrophic shallow water equations, Hassainia, Hmidi and Masmoudi [34] for generalized surface quasi-geostrophic equations, Roulley [45] for Euler- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} flows, Hassainia and Roulley [36] for Euler equations in the unit disk close to Rankine vortices and Hassainia, Hmidi and Roulley [35] for 2D Euler annular vortex patches. Time quasi-periodic solutions were also constructed for the 3D Euler equations with time quasi-periodic external force [7] and for the forced 2D Navier–Stokes equations [27] approaching in the zero viscosity limit time quasi-periodic solutions of the 2D Euler equations for all times.

We also mention that time quasi-periodic solutions for the Euler equations were constructed also by Crouseilles and Faou [17] in 2D, and by Enciso, Peralta-Salas and de Lizaur [22] in 3D and even dimensions: we remark that these latter solutions are engineered so that there are no small divisors issues to deal with, with consequently much easier proofs and a drawback of not having information on the potential stability of the solutions. Their construction has been recently adapted in [28] to prove the existence of time almost-periodic solutions for the Euler equation in 3D and even dimensions.

We conclude this part of the introduction with the following remarks. Our result, in companion with the recent work [16] on bi-periodic traveling wave solution for the non-resistive MHD system, is the first one in which the existence of large amplitude quasi-periodic solutions for a PDE with a forcing term of large size is proved. Up to now, large amplitude quasi-periodic solutions were constructed only for small perturbations of defocusing NLS and KdV equations by Berti, Kappeler and Montalto [12, 13], where the integrable structures of these two equations is exploited. We also mention that KAM techniques have been developed to study the dynamics of linear wave and Klein Gordon equations with potential of size O(1) and large frequencies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , see [24, 26].

Strategy of the proof

The solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\lambda }(\lambda \,\omega t,x)$$\end{document} in Theorem 1.3 that we are going to construct is a quasi-periodic traveling wave, according to Definition 1.2, of the form, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta >0$$\end{document} to determine,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v_{\lambda }({\varphi },x) = \lambda ^{\theta } w_{\lambda }({\varphi },x) , \quad w_{\lambda }({\varphi },x)=v_{\textrm{app},N} ({\varphi },x) + g({\varphi },x), \end{aligned}$$\end{document}

so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\lambda = O(\lambda ^{\theta })$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{\lambda } = O(1)$$\end{document} with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} . According to this rescaling, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{\lambda }({\varphi },x)$$\end{document} is searched as a quasi-periodic traveling solution of the equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big (\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )w_{\lambda } +\lambda ^{\theta } u \cdot \nabla w_{\lambda } = \lambda ^{\alpha -\theta } f(\lambda \, \omega t,x) , \quad u = {\mathfrak {B}}[w_{\lambda }]. \end{aligned}$$\end{document}

The leading order of function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{\lambda }({\varphi },x)$$\end{document} is the first approximation given by the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} of size O(1). The function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\varphi },x)$$\end{document} , on the other hand, will be a correction to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)$$\end{document} , bifurcating from the latter, of the smaller size o(1) with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} . If we follow the heuristic as in (1.13), we would expect to simply fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha -1$$\end{document} . However, due to the issue of small divisors in inverting the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} +\beta \,{\mathtt L}$$\end{document} , we have to worsen the size and take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} slightly larger than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha -1$$\end{document} to get an effective upper bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\lambda $$\end{document} of size O(1). We now sketchily illustrate how to construct these two terms and the main ideas of the strategy.

Construction of the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} . In order to retrieve a perturbative framework when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , the first step is to search for a function that solves (1.14) up to some error term that is perturbatively small, namely of size o(1) with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} . We do so by searching for a quasi-periodic traveling wave function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)$$\end{document} of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v_{\textrm{app},N}({\varphi },x) = v_0({\varphi },x) + v_{1}({\varphi },x) +...+ v_{N}({\varphi },x), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}_{0}$$\end{document} is the number of iterations needed to produce the desired correction to the approximation. The first term of this finite expansion is given as the solution of the linear equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big (\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big ) v_{0} = \lambda ^{\alpha -\theta } f . \end{aligned}$$\end{document}

Building upon the preceding discussion below Theorem 1.3, in order to mitigate the otherwise infinite-dimensional degeneracy in the kernel, the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}$$\end{document} has to be inverted on quasi-periodic traveling waves with zero average. But, to deal with the presence of small divisors, we need to impose a non-resonance condition with a consequent loss of derivatives in the action of the inverse. This non-resonance condition is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\lambda \,\omega \cdot \ell + \beta \, {\mathtt L}(j)| \geqslant \lambda \frac{\gamma }{|\ell |^\tau } , \quad \forall \,(\ell ,j) \in {\mathbb {Z}}^{\nu +2}\setminus \{0\} , \quad \pi ^\top (\ell )+j=0. \end{aligned}$$\end{document}

for some given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >\nu +1$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell )+j=0$$\end{document} is the Fourier restriction on the modes of quasi-periodic traveling waves. For this reason, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \big ( \lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )^{-1} \Vert _{{\mathcal {L}}(H^{s+\tau }_0,H^s_0)} \leqslant O(\lambda ^{-1}\gamma ^{-1}). \end{aligned}$$\end{document}

To impose smallness conditions later on, the small parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} coming from the non-resonance conditions will be linked to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} by choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\lambda ^{-{\mathtt c}}$$\end{document} , for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}\in (0,1)$$\end{document} sufficiently small. This implies that the upper bound in (1.16) is actually of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{-1+{\mathtt c}})$$\end{document} . On the other hand, together with the easy upper bound on the denominator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\lambda \,\omega \cdot \ell + \beta \,{\mathtt L}(j)| \leqslant \lambda |\omega ||\ell |$$\end{document} , under the assumption that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \geqslant |\beta |$$\end{document} and recalling (1.5), we will show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert \big ( \lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )^{-1} \Vert _{{\mathcal {L}}(H^{s}_0,H^{s+1}_0)} \geqslant O(\lambda ^{-1})$$\end{document} . Therefore, we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} O(\lambda ^{\alpha -\theta -1}) \leqslant \Vert v_{0}:= \big (\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )^{-1}[\lambda ^{\alpha -\theta } f] \Vert _{s} \leqslant O(\lambda ^{\alpha -\theta -1+{\mathtt c}}) . \end{aligned}$$\end{document}

To get the upper bound in (1.17) of size O(1), we fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha -1+{\mathtt c}$$\end{document} , whereas the lower bound becomes of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{-{\mathtt c}})$$\end{document} . These estimates of the leading order term of the solutions have the same rate, after rescaling, of the estimates of the whole solution in (1.12). For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=0$$\end{document} , the error that is produced by approximately solving (1.14) is given by the nonlinear term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta } {\mathfrak {B}}[v_0] \cdot \nabla v_0$$\end{document} , which is of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{\theta })=O(\lambda ^{\alpha -(1-{\mathtt c})})$$\end{document} , smaller than the size of the forcing term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\alpha -\theta } f= \lambda ^{1-{\mathtt c}}f$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}>0$$\end{document} small enough so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <2(1-{\mathtt c})$$\end{document} . The idea is to iterate this argument to construct the next-order approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_1$$\end{document} , which solves the linear equation with a forcing term represented by the error from the previous step. That is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v_1:= - \big ( \lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )^{-1} [\lambda ^{\theta } {\mathfrak {B}}[v_0] \cdot \nabla v_0] = O(\lambda ^{\theta -1+{\mathtt c}}) , \end{aligned}$$\end{document}

which is small with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} , so that the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},1}=v_0+v_1$$\end{document} solves (1.14) up to another error term that is even smaller with respect to the one created by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_0$$\end{document} alone. Iterating this procedure for N steps, we obtain the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N} = v_0 + v_1 + \ldots + v_N$$\end{document} , where each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots , N$$\end{document} , is small as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} in (1.15) satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big (\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big )v_{\textrm{app},N}+\lambda ^{\alpha -1} {\mathfrak {B}}[v_{\textrm{app},N}] \cdot \nabla v_{\textrm{app},N} - \lambda f({\varphi },x) = q_{N}({\varphi },x) , \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_N$$\end{document} a quasi-periodic traveling wave such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert q_N \Vert _{s} \leqslant O(\lambda ^{N(\alpha -2(1-{\mathtt c}))+\alpha -(1-{\mathtt c})})$$\end{document} . Choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N > \frac{\alpha -(1-{\mathtt c})}{2(1-{\mathtt c})-\alpha }$$\end{document} ensures \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert q_N\Vert _s \lesssim o(1)$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} . The detailed construction is provided in Proposition 3.3.

Linearized operator and Nash-Moser iteration. We are now in place to search for solution for (1.14) by means of a Nash-Moser iteration scheme, that is we look for the zeroes of the nonlinear functional

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {F}}(w):=\big (\lambda \,\omega \cdot \partial _{{\varphi }}+ \beta \,{\mathtt L}\big ) w +\lambda ^{\theta } {\mathfrak {B}}[w] \cdot \nabla w - \lambda ^{\alpha -\theta } f({\varphi },x) , \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta :=\alpha -1+{\mathtt c}>0$$\end{document} , as fixed before. We use as initial guess the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} constructed before and we want to find a sequence of quasi-periodic traveling wave functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(w_n = v_{\textrm{app},N} + g_n)_{n\in {\mathbb {N}}_{0}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_0:=0$$\end{document} , that converge to a zero of the nonlinear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document} . Roughly speaking, the sequence is determined via a Newton scheme, with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_{n+1} - w_{n} = - \big (\textrm{d}_{w} {\mathcal {F}}(w_n) \big )^{-1}\big [ {\mathcal {F}}(w_n) \big ]. \end{aligned}$$\end{document}

The main issue, therefore, is to analyze the linearized operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textrm{d}_{w} {\mathcal {F}}(w_n)$$\end{document} , determine its invertibility and estimate the inverse operator. Linearizing (1.18) at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n$$\end{document} , we focus on

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}:= \textrm{d}_{w} {\mathcal {F}}(w_n):= \lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ \lambda ^{\theta } {\mathfrak {B}}[w_n] \cdot \nabla + \lambda ^{\theta } \nabla w_n \cdot {\mathfrak {B}}. \end{aligned}$$\end{document}

The main issue is that, in the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}$$\end{document} in (1.19) is a perturbation of large size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^\theta )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta < 1$$\end{document} of the diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}$$\end{document} . The strategy to establish invertibility and related estimates for the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}$$\end{document} in (1.19) involves reducing it to a diagonal operator with respect to the Fourier basis. This reduction procedure will be carried out in three steps.

1. Reduction of the transport.
2. Reduction to smoothing and perturbative of the large remainder.
3. KAM reducibility scheme. If we achieve this reduction with a bounded invertible transformation close to the identity, we can deduce tame estimates for the action of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big (\textrm{d}_{w} {\mathcal {F}}(w_n) \big )^{-1}$$\end{document} . This operator will also experience a loss of derivatives, akin to the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}$$\end{document} . However, the Nash-Moser scheme compensates for this loss, ensuring fast convergence of the iterations in a low regularity norm while allowing controlled divergence in a high regularity norm. The invertibility of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}$$\end{document} in (1.19) is discussed in Sect. 6 after its reduction, while the Nash-Moser iteration is proven in Sect. 7 and finally Theorem 1.3 in Sect. 8.

The remaining part of this introduction will briefly describe the steps involved in the reduction scheme of Sect. 5, which constitutes a substantial part of this paper.

1) Reduction of the transport. The first step is to reduce the highest order term in (1.19), which is represented by the transport operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {T}}:= \lambda \,\omega \cdot \partial _{{\varphi }} + \lambda ^{\theta } {\mathfrak {B}}[w_n] \cdot \nabla = \lambda \big ( \omega \cdot \partial _{{\varphi }} + \lambda ^{\theta -1} {\mathfrak {B}}[w_n] \cdot \nabla \big ). \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta -1=\alpha -2+{\mathtt c}<0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < 2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}>0$$\end{document} small enough, the rescaled vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} {\mathfrak {B}}[w_n]$$\end{document} is perturbatively small with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} . Moreover, it has zero average in space and it is divergence free. Thanks to these properties, we will conjugate the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document} to fully reduce it to the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }}$$\end{document} , as soon as the frequency vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathbb {R}}^\nu \setminus \{0\}$$\end{document} is Diophantine (see (3.5)). This is the content of Proposition 5.3, which follows the scheme in [7].

The remaining terms in (1.19), namely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \,{\mathtt L}+ \lambda ^{\theta } \nabla w_n\cdot {\mathfrak {B}}$$\end{document} , undergo the same transformation to an operator of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \,{\mathtt L}+ {\mathcal {E}}_{1}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} is a remainder operator of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta }$$\end{document} belonging to the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} of operators that exhibit 1-smoothing according to off-diagonal matrix decay (see Definition (2.5) and subsequent properties for precise definitions). Additionally, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n({\varphi },x)$$\end{document} is a quasi-periodic traveling wave, the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} is momentum preserving, meaning it maps quasi-periodic traveling waves into quasi-periodic traveling waves (refer to Sect. 2.3 for definitions and characterizations). This property is crucial for the next two steps.

2) Reduction to a small and smoothing remainder. After the previous transformation, we are left to work with the operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{1}:= \lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ {\mathcal {E}}_{1}. \end{aligned}$$\end{document}

For the KAM reduction, the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} lacks necessary smoothing for derivative compensation and remains large for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} . The next step is to conjugate the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} to reduce both the size and order of the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} . This situation represents a novel development in normal form techniques compared to the previous literature.

To address this, we conjugate the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} in a series of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M-1$$\end{document} iterations, culminating in the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{M}$$\end{document} of the last iteration being of size o(1) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} and in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-M}$$\end{document} . We briefly descrive the first iteration. We conjugate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{1}= \textrm{exp} ({\mathcal {X}}_{1})$$\end{document} , with this transformation being invertible for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}_{1}={\mathcal {X}}_{1}({\varphi }) \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} sufficiently small. By expanding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{1}^{-1} {\mathcal {L}}_{1} \Phi _{1}$$\end{document} via the standard Lie expansion, the only term of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 1$$\end{document} is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \,\omega \cdot \partial _{{\varphi }} {\mathcal {X}}_{1} ({\varphi })+ {\mathcal {E}}_{1}({\varphi }). \end{aligned}$$\end{document}

The sublinear nature of the dispersion relation is crucial. Indeed the effect of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }}$$\end{document} is stronger than the dispersive operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \, {\mathtt L}$$\end{document} . This fact implies that the commutator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\mathcal {X}}_{1} ({\varphi }),\, \beta \,{\mathtt L}]$$\end{document} is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- 2$$\end{document} and hence it contributes to the remainder of the Lie expansion. We solve the homological equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \,\omega \cdot \partial _{{\varphi }} {\mathcal {X}}_{1} ({\varphi })+ {\mathcal {E}}_{1}({\varphi }) = {\widehat{{\mathcal {E}}}}_{1}(0) , \quad {\mathcal {E}}_{1}({\varphi }) = \sum _{\ell \in {\mathbb {Z}}^\nu } {\widehat{{\mathcal {E}}}}_{1}(\ell ) e^{\textrm{i}\ell \cdot {\varphi }} \end{aligned}$$\end{document}

in which we remove the dependence on time from the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}({\varphi })$$\end{document} by inverting the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }}$$\end{document} , provided the frequency vector is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} is Diophantine. A second key property here is that, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} is a momentum-preserving operator, the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\mathcal {E}}}}_{1}(0)$$\end{document} , that is the average in time of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}({\varphi })$$\end{document} , is diagonal with respect to the Fourier basis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(e^{\textrm{i}j \cdot x})_{j\in {\mathbb {Z}}^2}$$\end{document} (as shown in Lemma 2.13). We obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}_{1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} is of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{\theta -1+{\mathtt c}}) = O(1)$$\end{document} , as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} , ensuring estimates on the maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{1}^{\pm 1}$$\end{document} uniform with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough.

The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\mathcal {E}}}}_{1}(0)$$\end{document} contributes to a new diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ {\widehat{{\mathcal {E}}}}_{1}(0)$$\end{document} . The new remainder, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{2}$$\end{document} , contains terms that are neither solved by the homological equation (1.20) nor part of the normal form. The leading term of this new remainder is given by the commutator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\mathcal {X}}_{1},{\mathcal {E}}_{1}]$$\end{document} , which is of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{2\theta -1+{\mathtt c}})$$\end{document} and, by algebraic properties, in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-2}$$\end{document} . Therefore, the ensuing remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{2}$$\end{document} is smaller in size for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha -2(1-{\mathtt c})<0$$\end{document} (we then take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \mathtt c \ll 1$$\end{document} small enough) and exhibits more smoothing than the previous remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} . As in the construction of the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} , we iterate this procedure to reduce the remainder to a perturbatively small size. The details of this reduction are provided in Proposition 5.6.

3) KAM perturbative reduction. The conclusion of the previous step is the linear operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{M} = \lambda \,\omega \cdot \partial _{{\varphi }} + \textbf{D}_0 + \textbf{E}_0, \quad \textbf{D}_0:= \beta \,{\mathtt L}+ {\mathcal {Z}}_{M}, \quad \textbf{E}_0:= {\mathcal {E}}_{M}. \end{aligned}$$\end{document}

Here, the momentum preserving and constant coefficients operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ {\mathcal {Z}}_{M} $$\end{document} is diagonal with respect to the Fourier basis, featuring eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{i}\,\lambda \, \omega \cdot \ell + \mu _{0}(j)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{0}(j):=\textrm{i}\,\beta \, {\mathtt L}(j) + {\mathtt z}_{0,M}(j)\in {\mathbb {C}}$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu \setminus \{0\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in {\mathbb {Z}}^2\setminus \{0\}$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{M}\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-M}$$\end{document} is small, with a size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{M(\theta -1+{\mathtt c})+1-{\mathtt c}})$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} (note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta -1+{\mathtt c}< 0$$\end{document} ). If we assume parity properties on the external forcing term and we preserve the reversible structure of the system in all the steps of the reduction, than we get the diagonal elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt z}_{0,M}(j)$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}_{M}$$\end{document} are purely imaginary. Now, the goal is to fully reduce the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{M}$$\end{document} to a constant-coefficients diagonal operator through a series of conjugations. This process progressively annihilates the size of the perturbative remainder, ultimately yielding the final normal form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ \textbf{Z}_{\infty }$$\end{document} . A crucial problem, typical in PDEs in higher space dimension, is the presence of strong resonance phenomena. In this case, the difference of unperturbed eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt L}(j) - {\mathtt L}(j')$$\end{document} is zero for infinitely many \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} . Also at this step, this degeneracy issue is overcome by using the conservation of momentum. Indeed, at the first iterative step, the non-resonance conditions, required in the iterative KAM procedure, namey the second-order Melnikov conditions, take the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\textrm{i}\lambda \,\omega \cdot \ell + \mu _{0}(j)-\mu _{0}(j') | \geqslant \lambda \frac{\gamma }{{\langle {\ell }\rangle }^\tau |j'|^\tau } , \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu \setminus \{0\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j,j'\in {\mathbb {Z}}^2\setminus \{0\}$$\end{document} . It is noteworthy that the non-resonance conditions in (1.21), needed for estimating small divisors in the solution of the homological equations, lose regularity in both time and space. However, the key feature is the gain in size by a factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} . This is crucial to impose the appropriate smallness condition, ensuring the convergence of the scheme.

By imposing the non-resonance condition (1.21) above, one can construct a map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _0$$\end{document} solving the homological equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \, \omega \cdot \partial _{\varphi }\Psi _0({\varphi }) + [\textbf{D}_0,\, \Psi _0({\varphi })] + \textbf{E}_0({\varphi }) = \widehat{\textbf{E}}_0(0), \quad \widehat{\textbf{E}}_0(0) = \int _{{\mathbb {T}}^\nu } \textbf{E}_0({\varphi })\, \,\textrm{d}{{\varphi }}, \end{aligned}$$\end{document}

and one gets

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} e^{- \Psi _1} {{\mathcal {L}}}_M e^{\Psi _1} = \lambda \, \omega \cdot \partial _{\varphi }+ \textbf{D}_0 + \widehat{\textbf{E}}_0(0) + \textbf{E}_1, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_1$$\end{document} is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- M$$\end{document} and it has size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_1 \simeq \textbf{E}_0^2$$\end{document} which is essentially quadratic with respect to the size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_0$$\end{document} . It is now crucial to use the conservation of momentum, which allows to deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\textbf{E}}_0(0)$$\end{document} and also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_1: = \textbf{D}_0 + \widehat{\textbf{E}}_0(0)$$\end{document} are diagonal operators with respect to the Fourier basis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ e^{\textrm{i}j \cdot x}: j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \} \}$$\end{document} .

The detailed construction of the KAM reduction are provided in Sects. 5.3 and 5.4. As remarked below Theorem 1.3, if we assume the preservation of the reversible structure, then we can infer that the final constant-coefficients diagonal normal form has purely imaginary eigenvalues and deduce the linear stability of the final solution. Nevertheless,

the construction of these large amplitude solutions in Theorem 1.3 does not depend on this property.

Function Spaces, Norms and Linear Operators

Function spaces

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2,3$$\end{document} be the space dimension. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a: {\mathbb {T}}^\nu \times {\mathbb {T}}^d \rightarrow E$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a = a({\varphi },x)$$\end{document} , be a function, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E = {\mathbb {C}}^m$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^m$$\end{document} . Then, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in {\mathbb {R}}$$\end{document} , its Sobolev norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert a \Vert _s$$\end{document} is defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert a \Vert _s^2:= \sum _{(\ell , j) \in {\mathbb {Z}}^\nu \times {\mathbb {Z}}^d} \langle \ell , j \rangle ^{2s} | {\widehat{a}}(\ell ,j) |^2 , \quad \ \langle \ell , j \rangle := \max \{ 1, |\ell |, |j| \} , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{a}}(\ell ,j)$$\end{document} (either scalars or vectors) are the Fourier coefficients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a({\varphi },x)$$\end{document} , namely

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{a}}(\ell ,j):= \frac{1}{(2\pi )^{\nu +d}} \int _{{\mathbb {T}}^{\nu +d}} a({\varphi },x) e^{- \textrm{i}(\ell \cdot {\varphi }+ j \cdot x)} \, \,\textrm{d}{{\varphi }}\,\textrm{d}{x} . \end{aligned}$$\end{document}

We denote, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E = {\mathbb {C}}^m$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^m$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} H^s&:= H^s_{{\varphi },x}:= H^s({\mathbb {T}}^{\nu } \times {\mathbb {T}}^d) \\&:= H^s({\mathbb {T}}^{\nu } \times {\mathbb {T}}^d, E):= \{ u: {\mathbb {T}}^{\nu } \times {\mathbb {T}}^d \rightarrow E, \ \Vert u \Vert _s < \infty \} . \end{aligned} \end{aligned}$$\end{document}

In this paper, we use Sobolev norms for (real or complex, scalar- or vector-valued) functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u( {\varphi }, x; \omega )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x) \in {\mathbb {T}}^\nu \times {\mathbb {T}}^d$$\end{document} , being Lipschitz continuous with respect to the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathbb {R}}^\nu \setminus \{0\}$$\end{document} . We fix the threshold regularity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_0 \geqslant \nu +d + 3 \end{aligned}$$\end{document}

and we define the weighted Sobolev norms in the following way.

Definition 2.1

(Weighted Sobolev norms). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subseteq {\mathbb {R}}^{\nu }$$\end{document} be an arbitrary closed set

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0$$\end{document} as in (2.1). Given a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u: \Lambda \rightarrow H^s({\mathbb {T}}^\nu \times {\mathbb {T}}^d)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \mapsto u(\omega ) = u({\varphi },x; \omega )$$\end{document} that is Lipschitz continuous with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} , we define its weighted Sobolev norm by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u \Vert _{s}^{\textrm{Lip}(\gamma )}:= \Vert u \Vert _{s,\Lambda }^{\textrm{Lip}(\gamma )}:= \Vert u\Vert _{s}^{\sup } + \gamma \,\Vert u\Vert _{s-1}^\textrm{lip}, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert u\Vert _{s}^{\sup }: =\Vert u\Vert _{s,\Lambda }^{\sup }:= \sup _{\omega \in \Lambda } \Vert u(\omega )\Vert _{s}, \quad \Vert u\Vert _{s}^\textrm{lip}:= \Vert u\Vert _{s,\Lambda }^\textrm{lip}:= \sup _{\omega _1,\omega _2\in \Lambda \atop \omega _1\ne \omega _2} \frac{\Vert u(\omega _1)-u(\omega _2)\Vert _{s}}{| \omega _1-\omega _2|}. \end{aligned}$$\end{document}

For u independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} , we simply denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$| u |^{\textrm{Lip}(\gamma )}:= | u|^{\sup } + \gamma \, | u|^\textrm{lip} $$\end{document} .

Lemma 2.2

(Product). For all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ s \geqslant s_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert uv \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _s C(s) \Vert u \Vert _{s}^{\textrm{Lip}(\gamma )} \Vert v \Vert _{s_0}^{\textrm{Lip}(\gamma )}+ C(s_0) \Vert u \Vert _{s_0}^{\textrm{Lip}(\gamma )} \Vert v \Vert _{s}^{\textrm{Lip}(\gamma )}\,. \end{aligned}$$\end{document}

Notation. We will have also a dependence on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^{\nu }\ni \varphi \rightarrow w(\varphi , \cdot )\in L^2_0({\mathbb {T}}^2)$$\end{document} of which we want to control the regularity for the purposes of the nonlinear scheme. In particular, we will ask to be Lipschitz with respect to the function w as a parameter and therefore it is natural to introduce the following quantity. Given the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \mapsto g(w)$$\end{document} where g is an operator (or a map or a scalar function), we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _{12} g:= g(w_{2}) - g(w_{1}). \end{aligned}$$\end{document}

Quasi-periodic traveling functions

We restate the definition of quasi-periodic traveling as given in Definition 1.2, for functions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x)$$\end{document} instead of (t, x).

Definition 2.3

(Quasi-periodic traveling waves). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\jmath }}_{1},...,{\overline{\jmath }}_{\nu }\in {\mathbb {Z}}^d$$\end{document} be a given choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} vectors in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^d$$\end{document} . A function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:{\mathbb {T}}^\nu \times {\mathbb {T}}^d\rightarrow {\mathbb {R}}$$\end{document} is a quasi-periodic traveling wave if there exists a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{u}:{\mathbb {T}}^\nu \rightarrow {\mathbb {R}}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u({\varphi },x) = \breve{u}({\varphi }- \pi ( x)) , \quad \forall ({\varphi },x) \in {\mathbb {T}}^\nu \times {\mathbb {T}}^d, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^\nu $$\end{document} is the linear map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto \pi ( x):= (\overline{\jmath }_{1}\cdot x,...,\overline{\jmath }_{\nu }\cdot x)$$\end{document} .

Comparing with Definition 1.2 (with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} ), it is convenient to call quasi-periodic traveling wave both the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u({\varphi },x) = \breve{u}({\varphi }- \pi ( x))$$\end{document} and the function of time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(\omega t,x) = \breve{u}(\omega t- \pi ( x))$$\end{document} . We define the translation operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau _\varsigma : h(x) \mapsto h(x + \varsigma ),\;\;\varsigma \in {\mathbb {R}}^{d}. \end{aligned}$$\end{document}

Then, quasi-periodic traveling waves are also characterized by the relation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v(\varphi -\pi (\varsigma ),x)= ( \tau _\varsigma \circ v)(\varphi ,x)=v(\varphi ,x+\varsigma ), \quad \forall \,\varphi \in {\mathbb {T}}^\nu ,\; \varsigma \in {\mathbb {R}}^{d}, \; x\in {\mathbb {T}}^{d}. \end{aligned}$$\end{document}

Expanding in Fourier the equivalence in (2.4), we obtain that a quasi-periodic traveling wave has the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u({\varphi },x) = \sum _{{\begin{array}{c} (\ell , j) \in {\mathbb {Z}}^\nu \times {\mathbb {Z}}^d \\ \pi ^\top (\ell ) + j = 0 \end{array}}} {{\widehat{u}}}(\ell ,j) e^{\textrm{i}(\ell \cdot {\varphi }+ j\cdot x)}. \end{aligned}$$\end{document}

We define the subspace, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E= {\mathbb {C}}^m$$\end{document} or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^m$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_{\pi }:=\big \{v(\varphi , x)\in L^2({\mathbb {T}}^{\nu +d},E) \,: \, v(\varphi , x)=V(\varphi - \pi ( x)), \ V(\theta )\in L^2({\mathbb {T}}^\nu , E) \big \}. \end{aligned}$$\end{document}

With abuse of notation, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\pi }$$\end{document} also the subspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {T}}^{\nu +d}, {\mathbb {R}})$$\end{document} of scalar functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\varphi , x)$$\end{document} of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\varphi , x)=v(\varphi - \pi ( x)),\, V(\theta )\in L^2({\mathbb {T}}^\nu , {\mathbb {R}})$$\end{document} ). We note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (2.1.1) \quad \Leftrightarrow \quad v\in S_{\pi } \quad \Leftrightarrow \quad \pi ^\top \partial _{\varphi }v+\nabla v=0,\;\;\forall (\varphi ,x)\in {\mathbb {T}}^{\nu +d}. \end{aligned}$$\end{document}

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt K}\geqslant 1 $$\end{document} , we define the smoothing operators on quasi-periodic traveling waves of the form (2.5) as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} (\Pi _{{\mathtt K}} u)({\varphi },x):= \sum _{{\begin{array}{c} {\langle {\ell }\rangle }\leqslant {\mathtt K},\, j \in {\mathbb {Z}}^d \\ \pi ^\top (\ell ) + j = 0 \end{array}}} {{\widehat{u}}}(\ell ,j) e^{\textrm{i}(\ell \cdot {\varphi }+ j \cdot x)} , \quad \ \Pi ^\perp _{{\mathtt K}}:= \textrm{Id} - \Pi _{{\mathtt K}} . \end{aligned} \end{aligned}$$\end{document}

Note that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u({\varphi },x)$$\end{document} is a quasi-periodic traveling wave, then both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Pi _{{\mathtt K}} u)({\varphi },x)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Pi _{{\mathtt K}}^\perp u)({\varphi },x)$$\end{document} are quasi-periodic traveling waves as well.

Lemma 2.4

(Smoothing). The smoothing operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{{\mathtt K}}, \Pi _{{\mathtt K}}^\perp $$\end{document} satisfy the smoothing estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Pi _{{\mathtt K}} u \Vert _{s}^{\textrm{Lip}(\gamma )}&\leqslant {\mathtt K}^a \Vert u \Vert _{s-a}^{\textrm{Lip}(\gamma )}\, , \quad 0 \leqslant a \leqslant s \,, \\ \Vert \Pi _{{\mathtt K}}^\bot u \Vert _{s}^{\textrm{Lip}(\gamma )}&\leqslant {\mathtt K}^{-a} \Vert u \Vert _{s + a}^{\textrm{Lip}(\gamma )}\, , \quad a \geqslant 0 \,. \end{aligned}$$\end{document}

Matrix representation of linear operators

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}: L^2({\mathbb {T}}^d) \rightarrow L^2({\mathbb {T}}^d)$$\end{document} be a linear operator. It can be represented as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}} u (x):= \sum _{j, j' \in {\mathbb {Z}}^d} {\mathcal R}_j^{j'}{\widehat{u}}(j') e^{\textrm{i}j \cdot x} , \quad \text {for} \quad u (x) = \sum _{j \in {\mathbb {Z}}^d} {\widehat{u}}(j) e^{\textrm{i}j \cdot x} , \end{aligned}$$\end{document}

where, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^d$$\end{document} , the matrix element \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal R}_j^{j'}$$\end{document} is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}}_j^{j'}:= \frac{1}{(2\pi )^d} \int _{{\mathbb {T}}^d} {\mathcal R}[e^{\textrm{i}j' \cdot x}] e^{- \textrm{i}j \cdot x} \,\textrm{d}{x}. \end{aligned}$$\end{document}

We also consider smooth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} -dependent families of linear operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^\nu \rightarrow {{\mathcal {B}}} (L^2({\mathbb {T}}^d))$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \mapsto {{\mathcal {R}}}(\varphi )$$\end{document} , which we write in Fourier series with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}}(\varphi ) = \sum _{\ell \in {\mathbb {Z}}^\nu } \widehat{\mathcal R}(\ell ) e^{\textrm{i}\ell \cdot \varphi }, \quad \widehat{{\mathcal {R}}}(\ell ):= \frac{1}{(2 \pi )^\nu } \int _{{\mathbb {T}}^\nu } {{\mathcal {R}}}(\varphi ) e^{- \textrm{i}\ell \cdot \varphi }\, \,\textrm{d}{\varphi }, \quad \ell \in {\mathbb {Z}}^\nu . \end{aligned}$$\end{document}

According to (2.8), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^d$$\end{document} , the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathcal {R}}}(\ell ) \in {\mathcal B} (L^2({\mathbb {T}}^d))$$\end{document} is identified with the matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\widehat{\mathcal R}(\ell )_j^{j'})_{j, j' \in {\mathbb {Z}}^d}$$\end{document} . A map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^\nu \rightarrow {{\mathcal {B}}} (L^2({\mathbb {T}}^d))$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \mapsto {{\mathcal {R}}}(\varphi )$$\end{document} can be also regarded as a linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {T}}^{\nu +d}) \rightarrow L^2({\mathbb {T}}^{\nu +d })$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}} u(\varphi , x):= \sum _{{\begin{array}{c} \ell , \ell ' \in {\mathbb {Z}}^\nu \\ j, j' \in {\mathbb {Z}}^d \end{array}}} \widehat{{\mathcal {R}}}(\ell - \ell ')_j^{j '} {\widehat{u}}(\ell ', j') e^{\textrm{i}(\ell \cdot \varphi + j \cdot x)}, \quad \forall u \in L^2({\mathbb {T}}^{\nu + d}). \end{aligned}$$\end{document}

If the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}$$\end{document} is invariant on the space of functions with zero average in x, we identify \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}$$\end{document} with the matrix

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Big ( \widehat{{\mathcal {R}}}(\ell )_j^{j'} \Big )_{ j, j' \in {\mathbb {Z}}^d \setminus \{ 0 \}, \, \ell \in {\mathbb {Z}}^\nu } . \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}$$\end{document} be a linear operator as in (2.7)-(2.10). We define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal D}_{{\mathcal {R}}}$$\end{document} as the operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {D}}}_{{\mathcal {R}}}:= \textrm{diag}_{j \in {\mathbb {Z}}^2} \widehat{{\mathcal {R}}}(0)_j^j, \quad ({\mathcal {D}}_{{\mathcal {R}}})(\ell )_j^{j'}:= {\left\{ \begin{array}{ll} \widehat{{\mathcal {R}}}(\ell )_j^{j'} & \quad \text {if} \quad j=j', \ \ell =0, \\ 0 & \text {otherwise}. \end{array}\right. }. \end{aligned}$$\end{document}

In particular, we say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}$$\end{document} is a diagonal operator if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}\equiv {\mathcal {D}}_{{\mathcal {R}}}$$\end{document} .

For the purpose of the Normal Form methods for the linearized operator in Sect. 5, it is convenient to introduce the following norms. These norms take into account both the order and the off-diagonal decay of the matrix elements representing any linear operator on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {T}}^{\nu +d})$$\end{document} .

Definition 2.5

(Matrix decay norm and the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s$$\end{document} ). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in {\mathbb {R}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}$$\end{document} be an operator represented by the matrix in (2.10). We say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}$$\end{document} belongs to the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s$$\end{document} if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | {\mathcal {R}}|_{m, s}:= \sup _{j' \in {\mathbb {Z}}^d} \Big ( \sum _{(\ell ,j) \in {\mathbb {Z}}^{\nu +d}} \langle \ell , j-j' \rangle ^{2s} | \widehat{\mathcal {R}}(\ell )_j^{j'}|^2 \Big )^{\frac{1}{2}} \langle j' \rangle ^{- m} < \infty . \end{aligned}$$\end{document}

If the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}= {\mathcal {R}}(\omega )$$\end{document} is Lipschitz with respect to the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Lambda \subseteq {\mathbb {R}}^{\nu }$$\end{document} , we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&| {{\mathcal {R}}} |_{m, s}^{\textrm{Lip}(\gamma )}:= |{{\mathcal {R}}}|_{m, s}^\textrm{sup} + \gamma |{{\mathcal {R}}}|^\textrm{lip}_{m, s - 1}, \\&\quad |{{\mathcal {R}}}|_{m, s}^\textrm{sup}:= \sup _{\omega \in \Lambda } |{{\mathcal {R}}}(\omega )|_{m, s} , \quad |{{\mathcal {R}}}|_{m, s - 1}^\textrm{lip}:= \sup _{{\begin{array}{c} \omega _1, \omega _2 \in \Lambda \\ \omega _1 \ne \omega _2 \end{array}}} \dfrac{|{{\mathcal {R}}}(\omega _1) - {{\mathcal {R}}}(\omega _2)|_{m, s - 1}}{|\omega _1 - \omega _2|} . \end{aligned} \end{aligned}$$\end{document}

It readily follows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&m \leqslant m' \Longrightarrow {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s \subseteq {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{m'}_s \quad \text {and} \quad |\cdot |_{m', s}^{\textrm{Lip}(\gamma )} \leqslant |\cdot |_{m, s}^{\textrm{Lip}(\gamma )}, \\&\quad s \leqslant s' \Longrightarrow {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_{s'} \subseteq {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s \quad \text {and} \quad | \cdot |_{m, s}^{\textrm{Lip}(\gamma )} \leqslant |\cdot |_{m, s'}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

We now state some standard properties of the decay norms that are needed for the reducibility scheme of Sect. 5. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in H^s$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} , then the multiplication operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}_a: u \mapsto a u$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {M}}}_a \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^0 \quad \text {and} \quad |{\mathcal M}_a|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim \Vert a \Vert _s^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Lemma 2.6

(i) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^0_s$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u \Vert _s^{\textrm{Lip}(\gamma )} < \infty $$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {R}}} u \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _{s} |{\mathcal R}|_{0, s_0}^{\textrm{Lip}(\gamma )} \Vert u \Vert _s^{\textrm{Lip}(\gamma )} + |{{\mathcal {R}}}|_{0, s}^{\textrm{Lip}(\gamma )} \Vert u \Vert _{s_0}^{\textrm{Lip}(\gamma )}; \end{aligned}$$\end{document}

(ii) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m, m' \in {\mathbb {R}}$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Q}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{m'}_{s + |m|}$$\end{document} . Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} {{\mathcal {Q}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{m + m'}_s$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{{\mathcal {R}}}{{\mathcal {Q}}}|_{m + m', s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} |{{\mathcal {R}}}|_{m, s}^{\textrm{Lip}(\gamma )} |{\mathcal Q}|_{m', s_0 + |m|}^{\textrm{Lip}(\gamma )} + |{{\mathcal {R}}}|_{m, s_0}^{\textrm{Lip}(\gamma )} |{{\mathcal {Q}}}|_{m', s + |m|}^{\textrm{Lip}(\gamma )}\ ; \end{aligned}$$\end{document}

(iii) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\geqslant 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{-m}_s$$\end{document} . Then, for any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}^n \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{-m}_s$$\end{document} and there exist constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(s_0,m), C(s, m) > 0$$\end{document} , independent of n, such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {R}}}^n|_{-m, s_0}^{\textrm{Lip}(\gamma )} \leqslant C(s_0, m)^{n - 1} \big (|{{\mathcal {R}}}|_{-m, s_0}^{\textrm{Lip}(\gamma )}\big )^{n} , \\&\quad |{{\mathcal {R}}}^n|_{-m, s}^{\textrm{Lip}(\gamma )} \lesssim \, \big (C(s, m)|{{\mathcal {R}}}|_{-m, s_0}^{\textrm{Lip}(\gamma )}\big )^{n - 1} |{{\mathcal {R}}}|_{-m, s}^{\textrm{Lip}(\gamma )}; \end{aligned} \end{aligned}$$\end{document}

(iv) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \geqslant 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- m}_s$$\end{document} . Then there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (s,m) \in (0, 1)$$\end{document} small enough such that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {R}}}|_{-m, s_0}^{\textrm{Lip}(\gamma )} \leqslant \delta (s,m)$$\end{document} , then the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi := \textrm{exp}({{\mathcal {R}}}) \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{0}_s$$\end{document} is invertible and satisfies the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\Phi ^{\pm 1}|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _s 1 + |{\mathcal R}|_{-m, s}^{\textrm{Lip}(\gamma )}; \end{aligned}$$\end{document}

(v) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in {\mathbb {R}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}_{{\mathcal {R}}}$$\end{document} be the diagonal operator as in (2.11) Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}_{{\mathcal {R}}} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^m_s$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {D}}}_{{\mathcal {R}}}|_{m, s}^{\textrm{Lip}(\gamma )} \lesssim |{{\mathcal {R}}}|_{m, s_0}^{\textrm{Lip}(\gamma )}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} . As a consequence,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | \widehat{{\mathcal {R}}}(0)_j^j |^{\textrm{Lip}(\gamma )} \lesssim \langle j \rangle ^m|{{\mathcal {R}}}|_{s_0}^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Proof

Items (i), (ii) are proved in Lemma 2.6 in [28]. To prove item (iii) we need the following preliminary estimates:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {R}}}^n|_{0, s_0}^{\textrm{Lip}(\gamma )} \leqslant {\mathtt C}(s_0)^{n - 1} \big (|{{\mathcal {R}}}|_{0, s_0}^{\textrm{Lip}(\gamma )}\big )^{n} , \\&\quad |{{\mathcal {R}}}^n|_{0, s}^{\textrm{Lip}(\gamma )} \leqslant \big ({\mathtt C}(s_0)|{{\mathcal {R}}}|_{0, s_0}^{\textrm{Lip}(\gamma )}\big )^{n - 1} |{{\mathcal {R}}}|_{0, s}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

One can easily prove them by an induction argument and by using item (ii) (see also Lemma 2.6-(iii) in [28]). We now prove (2.14). By item (ii), (2.15) and (2.13), we compute, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |{\mathcal {R}}^{n-1} {\mathcal {R}}|_{-m, s_0}^{\textrm{Lip}(\gamma )}&\lesssim _{m,s_0} |{\mathcal {R}}^{n-1}|_{0, s_0}^{\textrm{Lip}(\gamma )} |{\mathcal {R}}|_{-m, s_0}^{\textrm{Lip}(\gamma )} \\&\lesssim _{m,s_0} ({\mathtt C}(s_0))^{n-2}(|{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{0,s_0})^{n-1}|{\mathcal {R}}|_{-m, s_0}^{\textrm{Lip}(\gamma )} \\&\leqslant c(m,s_0) ({\mathtt C}(s_0))^{n-2}(|{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{-m,s_0})^{n}, \\ |{\mathcal {R}}^{n-1} {\mathcal {R}}|_{-m, s}^{\textrm{Lip}(\gamma )}&\lesssim _{m,s} |{\mathcal {R}}^{n-1}|_{0, s}^{\textrm{Lip}(\gamma )} |{\mathcal {R}}|_{-m, s_0}^{\textrm{Lip}(\gamma )} + |{\mathcal {R}}^{n-1}|_{0, s_0}^{\textrm{Lip}(\gamma )} |{\mathcal {R}}|_{-m, s}^{\textrm{Lip}(\gamma )} \\&\lesssim _{m,s} ({\mathtt C}(s)|{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{0,s_0})^{n-2} |{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{0,s} |{\mathcal {R}}|_{-m, s_0}^{\textrm{Lip}(\gamma )} \\&\quad \, \,+ ({\mathtt C}(s_0))^{n-2}(|{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{0,s_0})^{n-1}|{\mathcal {R}}|_{-m, s}^{\textrm{Lip}(\gamma )} \\&\leqslant c(m,s) {\mathtt C}(s)^{n-2} (|{\mathcal {R}}|^{\textrm{Lip}(\gamma )}_{-m,s_0})^{n-1}|{\mathcal {R}}|_{-m, s}^{\textrm{Lip}(\gamma )} , \end{aligned} \end{aligned}$$\end{document}

and the bound follows provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(m,s_0)\geqslant \max \{ c(m,s_0), {\mathtt C}(s_0)\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(m,s)\geqslant \max \{ c(m,s), {\mathtt C}(s)\}$$\end{document} . Item (iv) follows by recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^{\pm 1} = \textrm{exp}(\pm {\mathcal {R}}) = \textrm{Id}+ \sum _{k=1}^{\infty } \frac{{{\mathcal {R}}}^{n}}{n!}$$\end{document} and by (2.15), (2.13). The claims of item (v) are a direct consequence of the definition of the matrix decay norm in Definition 2.12. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N > 0$$\end{document} , we define the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _N {{\mathcal {R}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _N^\perp {\mathcal {R}}$$\end{document} by means of their matrix representation as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\widehat{\Pi _N {{\mathcal {R}}}})(\ell )_{j}^{j'}:= {\left\{ \begin{array}{ll} \widehat{{\mathcal {R}}}(\ell )_j^{j'} & \text {if } |\ell |, |j - j'| \leqslant N, \\ 0 & \text {otherwise}, \end{array}\right. } \qquad \ \Pi _N^\bot {{\mathcal {R}}}:= {{\mathcal {R}}} - \Pi _N {{\mathcal {R}}}.\qquad \end{aligned}$$\end{document}

Lemma 2.7

For all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s, \alpha \geqslant 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in {\mathbb {R}}$$\end{document} , one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Pi _N {\mathcal R}|_{m, s + \alpha }^{\textrm{Lip}(\gamma )} \leqslant N^\alpha |{\mathcal R}|_{m, s}^{\textrm{Lip}(\gamma )}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Pi _N^\bot {{\mathcal {R}}}|_{m, s}^{\textrm{Lip}(\gamma )} \leqslant N^{- \alpha } |{{\mathcal {R}}}|_{m, s + \alpha }^{\textrm{Lip}(\gamma )}$$\end{document} .

Proof

The claims follow directly from (2.12) and (2.16). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We also define the projection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _0$$\end{document} on the space of zero average functions as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Pi _0 h:= \frac{1}{(2 \pi )^{ d}} \int _{{\mathbb {T}}^{ d}} h(\varphi , x)\, \,\textrm{d}{x} , \qquad \Pi _0^\bot := \textrm{Id} - \Pi _0. \end{aligned}$$\end{document}

In particular, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m, s \geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\Pi _0^\bot |_{0, s} \leqslant 1, \quad |\Pi _0|_{- m, s} \lesssim _{m} 1. \end{aligned}$$\end{document}

We finally mention the elementary properties of the Laplacian operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \Delta $$\end{document} and its inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(- \Delta )^{- 1}$$\end{document} acting on functions with zero average in x:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \Delta u(x) = \sum _{\xi \in {\mathbb {Z}}^d \setminus \{ 0 \}} |\xi |^2 {\widehat{u}}(\xi ) e^{\textrm{i}x \cdot \xi } , \quad (- \Delta )^{- 1} u(x) = \sum _{\xi \in {\mathbb {Z}}^d \setminus \{ 0 \}} \frac{1}{|\xi |^2} \widehat{u}(\xi ) e^{\textrm{i}x \cdot \xi }.\qquad \end{aligned}$$\end{document}

By Definition 2.5, one easily verifies, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |- \Delta |_{2, s} \leqslant 1, \quad |(- \Delta )^{- 1}|_{- 2, s} \leqslant 1. \end{aligned}$$\end{document}

Finally, recalling the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt L$$\end{document} in (1.5) we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | {\mathtt L}|_{-1, s} \leqslant 1. \end{aligned}$$\end{document}

Real and reversible operators

For any function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u({\varphi },x)$$\end{document} , we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in X$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u = \textrm{even}({\varphi },x)$$\end{document} , meaning that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u({\varphi },x)=u(-{\varphi },-x)$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in Y$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u = \textrm{odd}({\varphi },x)$$\end{document} , meaning that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(-{\varphi },-x)=-u({\varphi },x)$$\end{document} .

Definition 2.8

(i) We say that a linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is reversible if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : X \rightarrow Y$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : Y \rightarrow X$$\end{document} . We say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is reversibility preserving if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : X \rightarrow X$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : Y \rightarrow Y$$\end{document} .

(ii) We say that an operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : L^2({\mathbb {T}}^d) \rightarrow L^2({\mathbb {T}}^d)$$\end{document} is real if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (u)$$\end{document} is real valued for any u real valued.

It is convenient to reformulate real and reversibility properties of linear operators in terms of their matrix representations.

Lemma 2.9

A linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}$$\end{document} is:

(i) real if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathcal R}(\ell )_{j}^{j'} = \overline{\widehat{{\mathcal {R}}}(-\ell )_{- j}^{- j'}}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^d$$\end{document} ;

(ii) reversible if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathcal R}(\ell )_j^{j'} = - \widehat{{\mathcal {R}}}(-\ell )_{- j}^{- j'}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^d$$\end{document} ;

(iii) reversibility-preserving if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathcal {R}}}(\ell )_j^{j'} = \widehat{{\mathcal {R}}}(-\ell )_{- j}^{- j'}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^d$$\end{document} .

Momentum preserving operators

The following definition is crucial in the construction of traveling waves.

Definition 2.10

(Momentum preserving). A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi }$$\end{document} -dependent family of linear operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi }) $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi }\in {\mathbb {T}}^\nu $$\end{document} , is momentum preserving if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {R}}({\varphi }- \pi (\varsigma ) ) \circ \tau _\varsigma = \tau _\varsigma \circ {\mathcal {R}}({\varphi }) \,, \quad \forall \,{\varphi }\in {\mathbb {T}}^\nu \,, \ \varsigma \in {\mathbb {R}}^d \,, \end{aligned}$$\end{document}

where the translation operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _\varsigma $$\end{document} is defined in (2.3).

In particular, we mention that the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{-1}$$\end{document} defined in (2.19) are momentum preserving.

Momentum preserving operators are closed under several operations.

Lemma 2.11

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi }), {\mathcal {Q}}({\varphi })$$\end{document} be momentum preserving operators. Then:

• (i)(Composition): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi }) \circ {\mathcal {Q}}({\varphi }) $$\end{document} is a momentum preserving operator;
• (ii)(Inversion): If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi })$$\end{document} is invertible then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi })^{-1}$$\end{document} is momentum preserving;
• (iii)(Flow): Assume that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{t} \Phi ^t ({\varphi }) = {\mathcal {R}}({\varphi }) \Phi ^t ({\varphi }) \,, \quad \Phi ^0 ({\varphi }) = \textrm{Id} \,, \end{aligned}$$\end{document}

has a unique propagator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^t ({\varphi }) $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t\in [0,1] $$\end{document} . Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^t ( {\varphi }) $$\end{document} is momentum preserving.

Proof

Item (i) follows directly by (2.22). Item (ii) follows by taking the inverse, of (2.22) and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{-\varsigma } = \tau _\varsigma ^{-1} $$\end{document} . Finally, item (iii) holds because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tau _\varsigma ^{-1} \Phi ^t ( {\varphi }- \pi (\varsigma )) \tau _\varsigma $$\end{document} solves the same Cauchy problem in (2.23). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.12

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi })$$\end{document} be a momentum preserving linear operator and u a quasi-periodic traveling wave, according to Definition 2.3. Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi }) u $$\end{document} is a quasi-periodic traveling wave.

Proof

It follows by Definition 2.10 and by the characterization of traveling waves in (2.4). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now provide a characterization of the momentum preserving property in Fourier space. In particular, momentum preserving operators that are independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }\in {\mathbb {T}}^\nu $$\end{document} are actually diagonal.

Lemma 2.13

A \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi }$$\end{document} -dependent family of operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {R}}({\varphi }) $$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varphi }\in {\mathbb {T}}^\nu $$\end{document} , is momentum preserving if and only if the matrix elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi })$$\end{document} , defined by (2.10), fulfill

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{{\mathcal {R}}}}(\ell )_{j}^{j'} \ne 0 \quad \Rightarrow \quad \pi ^\top ( \ell ) + j-j' = 0 \,, \quad \forall \, \ell \in {\mathbb {Z}}^\nu , \ \ j,j'\in {\mathbb {Z}}^d \,. \end{aligned}$$\end{document}

As a consequence we have that, for any momentum preserving operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}({\varphi })$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathcal {R}}}(0)$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{{\mathcal {R}}}}(0)_{j}^{j'} \ne 0 \quad \Rightarrow \quad j=j' , \end{aligned}$$\end{document}

that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widehat{{\mathcal {R}}}}}(0)$$\end{document} is a time-independent diagonal operator (recall (2.11)).

Proof

By (2.10) we have, for any function u(x),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau _\varsigma ( {\mathcal {R}}({\varphi }) u ) = \sum _{j,j'\in {\mathbb {Z}}^d}\sum _{\ell \in {\mathbb {Z}}^\nu } {\widehat{{\mathcal {R}}}}(\ell )_j^{j'} e^{\textrm{i}j \cdot \varsigma } u_{j'} e^{\textrm{i}(\ell \cdot {\varphi }+ j\cdot x )} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {R}}({\varphi }-\pi (\varsigma )) [\tau _\varsigma u] = \sum _{j,j'\in {\mathbb {Z}}^d}\sum _{\ell \in {\mathbb {Z}}^\nu } {\widehat{{\mathcal {R}}}}(\ell )_j^{j'} e^{- \textrm{i}\ell \cdot \pi (\varsigma )} e^{\textrm{i}j' \cdot \varsigma } u_{j'} e^{\textrm{i}(\ell \cdot {\varphi }+ j\cdot x )} \,. \end{aligned}$$\end{document}

Therefore, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \cdot \pi (\varsigma ) = \pi ^\top (\ell )\cdot \varsigma $$\end{document} , we obtain that (2.22) is equivalent to (2.24). The claim in (2.25) follows immediately by (2.24): indeed, assuming that the matrix elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ( {\widehat{{\mathcal {R}}}}(0)_{j}^{j'}\big )_{j,j'\in {\mathbb {Z}}^d, \, \ell \in {\mathbb {Z}}^\nu }$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widehat{{\mathcal {R}}}}} (0)$$\end{document} are zero outside the diagonal, by using the momentum condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi ^\top ( \ell ) + j-j' = 0 $$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = 0$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Conjugations with change of variables

Here we recall some results from [7] regarding certain properties of linear operators with matrix decay that are conjugated by a change of variables

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {A}}={\mathcal {A}}({\varphi }): u \mapsto {\mathcal {A}}({\varphi }) u, \quad ({\mathcal {A}}({\varphi })u)(x):= u(x+{\varvec{\alpha }}({\varphi },x)), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}: {\mathbb {T}}^\nu \times {\mathbb {T}}^d \rightarrow {\mathbb {R}}^d$$\end{document} . We have the following lemmata.

Lemma 2.14

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\varvec{\alpha }}\Vert _{s_0}^{\textrm{Lip}(\gamma )}\leqslant \delta (s_0)$$\end{document} sufficiently small. The composition operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} in (2.26) satisfies the tame estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\mathcal {A}}u \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert u\Vert _{s}^{\textrm{Lip}(\gamma )} + \Vert {\varvec{\alpha }}\Vert _{s}^{\textrm{Lip}(\gamma )} \Vert u \Vert _{s_0}^{\textrm{Lip}(\gamma )} . \end{aligned}$$\end{document}

Moreover, it is an invertible map, with inverse given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {A}}^{-1}= {\mathcal {A}}({\varphi })^{-1}: u \mapsto {\mathcal {A}}({\varphi })^{-1} u , \quad ({\mathcal {A}}({\varphi })^{-1} u)(y):= u(y+\breve{{\varvec{\alpha }}}({\varphi },y)), \end{aligned}$$\end{document}

for some function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\alpha }}}({\varphi },y)$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x:=y+\breve{{\varvec{\alpha }}}({\varphi },y)$$\end{document} is the inverse diffeomorphism of the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y:=x+{\varvec{\alpha }}({\varphi },x)$$\end{document} , with estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \breve{{\varvec{\alpha }}}\Vert _{s}^{\textrm{Lip}(\gamma )}\lesssim _{s} \Vert {\varvec{\alpha }}\Vert _{s}^{\textrm{Lip}(\gamma )} , \quad \Vert {\mathcal {A}}^{-1} u \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert u\Vert _{s}^{\textrm{Lip}(\gamma )} + \Vert {\varvec{\alpha }}\Vert _{s}^{\textrm{Lip}(\gamma )} \Vert u \Vert _{s_0}^{\textrm{Lip}(\gamma )} , \\&\quad \Vert ({\mathcal {A}}^{\pm 1} -\textrm{Id})u \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert {\varvec{\alpha }}\Vert _{s_0+1}^{\textrm{Lip}(\gamma )} \Vert u\Vert _{s+1}^{\textrm{Lip}(\gamma )} + \Vert {\varvec{\alpha }}\Vert _{s+1}^{\textrm{Lip}(\gamma )} \Vert u \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

Proof

We refer to Lemma 2.3-(ii), (iii) in [7] for the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.15

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\varvec{\alpha }}({\varphi }, x ) $$\end{document} is a quasi-periodic traveling wave, then the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {A}}({\varphi })$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}({\varphi })^{-1}$$\end{document} defined in (2.26)-(2.27) are momentum preserving.

Proof

We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {A}}({\varphi }- \pi (\varsigma )) [\tau _\varsigma u]&= u(x+ {\varvec{\alpha }}({\varphi }- \pi (\varsigma ),x) + \varsigma ) \\&= u(x+ \varsigma + {\varvec{\alpha }}({\varphi },x+ \varsigma )) = \tau _\varsigma \big ( {\mathcal {A}}({\varphi }) u\big ) . \end{aligned} \end{aligned}$$\end{document}

This proves that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}({\varphi })$$\end{document} is momentum preserving. By Lemma 2.11-(ii), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}({\varphi })^{-1}$$\end{document} is momentum preserving as well. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.16

(Conjugation of the Laplacian). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S>s_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}\in H^{S+2}({\mathbb {T}}^{\nu +d})$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\varvec{\alpha }}\Vert _{s_0+\mu }^{\textrm{Lip}(\gamma )}\leqslant \delta $$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =\delta (d,\nu )\in (0,1)$$\end{document} small enough and for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =\mu (d,\nu )>0$$\end{document} . The following hold:

(i) The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta $$\end{document} acting on functions with zero average in x, as defined in (2.19), is conjugated via (2.26)-(2.27) to the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }$$\end{document} acting on functions with zero average in x defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{-\Delta }:= \Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta ) {\mathcal {A}}\Pi _0^\perp = -\Delta + {\mathcal {P}}_{2}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{2} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^2_s$$\end{document} satisfies, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{\mathcal {P}}_{2} |_{2,s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert {\varvec{\alpha }}\Vert _{s+\mu }^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Furthermore, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}({\varphi },x)$$\end{document} is a quasi-periodic traveling wave, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{2}$$\end{document} are momentum preserving, according to Definition 2.10;

(ii) The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }$$\end{document} in (2.28) is invertible on the functions of zero average in x, with inverse given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{-\Delta }^{-1}:= \Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta )^{-1} {\mathcal {A}}\Pi _0^\perp = (-\Delta )^{-1}+ {\mathcal {P}}_{-2}, \end{aligned}$$\end{document}

where the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{-1}$$\end{document} acting on functions with zero average in x is defined in (2.19), whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-2} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{-2}$$\end{document} satisfies, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{\mathcal {P}}_{-2} |_{-2,s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert {\varvec{\alpha }}\Vert _{s+\mu }^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Furthermore, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}({\varphi },x)$$\end{document} is a quasi-periodic traveling wave, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }^{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-2}$$\end{document} are momentum preserving, according to Definition 2.10.

Proof

We start with item (i). We compute explicitly

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {A}}^{-1} (-\Delta ) {\mathcal {A}}= -\Delta + \sum _{i,j=1}^{d} a_{ij}({\varphi },y) \partial _{y_i} \partial _{y_j} + \sum _{j=1}^{d} b_{j} ({\varphi },y) \partial _{y_j} , \end{aligned} \end{aligned}$$\end{document}

where, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i,j=1,...,d$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} a_{jj}({\varphi },y)&:= - {\mathcal {A}}^{-1} \big \{ | \nabla (x_j + \alpha _{j}({\varphi },x)) |^2 - 1 \big \}, \quad i= j , \\ a_{i,j}({\varphi },y)&:= - 2 {\mathcal {A}}^{-1}\big \{ \nabla (x_i + \alpha _{i}({\varphi },x)) \cdot \nabla (x_j + \alpha _{i}({\varphi },x)) \big \} , \quad i\ne j ,\\ b_{j}({\varphi },y)&:= - {\mathcal {A}}^{-1} \big \{ \Delta \alpha _{j}({\varphi },x) \big \}, \quad {\varvec{\alpha }}({\varphi },x) = (\alpha _{1}({\varphi },x),...,\alpha _{d}({\varphi },x)). \end{aligned} \end{aligned}$$\end{document}

We conclude that the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{2}$$\end{document} in (2.28) is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{2}:= \Pi _0^\perp \Big ( \sum _{i,j=1}^{d} a_{ij}({\varphi },y) \partial _{y_i} \partial _{y_j} + \sum _{j=1}^{d} b_{j} ({\varphi },y) \partial _{y_j} \Big ) , \end{aligned}$$\end{document}

and the estimate (2.29) follows by the explicit formulae of the coefficients in (2.32), Lemma 2.6-(ii) and (2.18). Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}({\varphi },x)$$\end{document} is a traveling wave, and since the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _0^\perp $$\end{document} are momentum preserving, it follows from Lemma  2.15 that the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{2}$$\end{document} are momentum preserving as well.

We now prove item (ii). First, note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {P}}_{-\Delta } {\mathcal {P}}_{-\Delta }^{-1}&= \Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta ) {\mathcal {A}}\Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta )^{-1} {\mathcal {A}}\Pi _0^\perp \\&= \Pi _0^\perp - \Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta ) {\mathcal {A}}\Pi _0 {\mathcal {A}}^{-1} (-\Delta )^{-1} {\mathcal {A}}\Pi _0^\perp \\&= \Pi _0^\perp - \Pi _0^\perp {\mathcal {A}}^{-1} (-\Delta ) \Pi _0 {\mathcal {A}}^{-1} (-\Delta )^{-1} {\mathcal {A}}\Pi _0^\perp = \Pi _0^\perp , \end{aligned} \end{aligned}$$\end{document}

since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}\Pi _0 = \Pi _0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )\Pi _0 = 0$$\end{document} , so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }^{-1}$$\end{document} defined in (2.30) is indeed the inverse of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }$$\end{document} on the space of function with zero average with respect to x. Next, we write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{-\Delta } = -\Delta + {\mathcal {P}}_{2} = (-\Delta ) \big ( \textrm{Id} + F \big ), \quad F:= (-\Delta )^{-1} {\mathcal {P}}_{2}, \end{aligned}$$\end{document}

where, by Lemma 2.6-(ii) and (2.20), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^0$$\end{document} and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | F|_{0,s}^{\textrm{Lip}(\gamma )} \lesssim _{s} | {\mathcal {P}}_{2}|_{2,s+2}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert {\varvec{\alpha }}\Vert _{s+\mu }^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Then, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\varvec{\alpha }}\Vert _{s_0 +\mu }^{\textrm{Lip}(\gamma )}$$\end{document} sufficiently small, the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Id}+ F \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^0$$\end{document} is invertible by a standard Neumann series argument, with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textrm{Id}+ F)^{-1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^0$$\end{document} . We obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{-\Delta }^{-1} = (\textrm{Id} + F)^{-1} (-\Delta )^{-1} = (-\Delta )^{-1} + {\mathcal {P}}_{-2} , \end{aligned}$$\end{document}

where, by Lemma 2.6 and (2.20),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {P}}_{-2}:= \big [ (\mathrm Id + F)^{-1} - \textrm{Id} \big ] (-\Delta )^{-1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-2}, \end{aligned}$$\end{document}

and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s \leqslant S$$\end{document} , satisfies the estimate (2.31). Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}({\varphi },x)$$\end{document} is a traveling wave, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-\Delta }^{-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{-2}$$\end{document} are momentum preserving by item (i), Lemma 2.11 and the fact that the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{-1}$$\end{document} is momentum preserving. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Construction of the Approximate Solution

The first step in the search for a solution of (1.9) is to find an approximate solution of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\lambda ^{\theta })$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha - 1< \theta < 1$$\end{document} to be determined. By rescaling the unknown \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \rightsquigarrow \lambda ^{\theta } v $$\end{document} in (1.9), we look for quasi-periodic traveling wave solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v({\varphi },x)$$\end{document} of size O(1) as zeroes of the nonlinear map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}(v) \equiv {{\mathcal {F}}}(v; \omega , \lambda )$$\end{document} of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{{\mathcal {F}}}(v):= \lambda \,\omega \cdot \partial _{{\varphi }} v + \beta \,{\mathtt L}v + \lambda ^{\theta } {{\mathcal {N}}}(v,v) - \lambda ^{\alpha -\theta } f({\varphi },x) =0, \\&\quad {{\mathcal {N}}}(v_1,v_2): = {\mathfrak {B}}[v_1] \cdot \nabla v_2. \end{aligned} \end{aligned}$$\end{document}

If we want to construct reversible solutions, we require the forcing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi },x)$$\end{document} to satisfy the symmetry condition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f({\varphi },x) = \textrm{even}({\varphi },x), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textrm{even}({\varphi },x)$$\end{document} denotes a function which is even in both variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }$$\end{document} and x. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\varphi , x)$$\end{document} a quasi-periodic traveling wave, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}} \equiv {{\mathcal {L}}}(v):= \textrm{d}_{v}{{\mathcal {F}}}(v)$$\end{document} the linearized operator near v:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{{\mathcal {L}}} = \lambda \, \omega \cdot \partial _\varphi + \beta \, {\mathtt L}+ \lambda ^{\theta } \big ( \textbf{a}_0(\varphi , x) \cdot \nabla + {{\mathcal {E}}}_0 \big ), \\&\quad \textbf{a}_0(\varphi , x): = {\mathfrak {B}}\big [ v \big ] (\varphi , x) , \quad {\mathcal {E}}_{0}[h]:= \nabla v \cdot {\mathfrak {B}}[h] , \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt L}$$\end{document} is defined in (1.5) and the Biot-Savart operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathfrak {B}}:= \nabla ^\perp (-\Delta )^{-1} $$\end{document} is as in (1.8).

From now on, the dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} is fixed once and for all, so that all the definitions and properties in Sect. 2 will be applied with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} .

First we establish some quantitative estimates on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}$$\end{document} that we shall use in the sequel and its behaviour when acting on quasi-periodic traveling waves.

Lemma 3.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0>0 $$\end{document} be as in (2.1). The following hold:

(i) Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s > s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\,\cdot \,; \omega ) \in H^{s + 1}_0({\mathbb {T}}^{\nu + 2})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v \Vert _{s_0 + 1}^{\textrm{Lip}(\gamma )} \lesssim 1$$\end{document} . Then the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}(v)$$\end{document} in (3.1) satisfies the following estimate:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {F}}}(v) \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _s \lambda \big (1 + \Vert v \Vert _{s + 1}^{\textrm{Lip}(\gamma )} \big ). \end{aligned}$$\end{document}

Moreover, if v is a quasi-periodic traveling wave, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}(v)$$\end{document} is a quasi-periodic traveling wave. In addition, assuming (3.2), if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}(v)({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} ;

(ii) Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v \Vert _{s_0 + 1}^{\textrm{Lip}(\gamma )} \lesssim 1$$\end{document} . Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\,\cdot \,;\omega ) \in H^{s_0 + b}_0({\mathbb {T}}^{\nu + 2})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt K}\in {\mathbb {N}}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {{\mathcal {L}}} h \Vert _{s_0}^{\textrm{Lip}(\gamma )} \lesssim \lambda \Vert h \Vert _{s_0 + 1}^{\textrm{Lip}(\gamma )}, \\&\quad \Vert [{\mathcal {L}},\Pi _{{\mathtt K}}^\perp ] h \Vert _{s_0}^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta } {\mathtt K}^{1-b}\big ( \Vert h \Vert _{s_0+b}^{\textrm{Lip}(\gamma )} + \Vert v \Vert _{s_0+b}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} \big ) , \\&\quad \Vert [\Pi _{{\mathtt K}},{\mathcal {L}}] h \Vert _{s_0+b}^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta } {\mathtt K}\big ( \Vert h \Vert _{s_0+b}^{\textrm{Lip}(\gamma )} + \Vert v \Vert _{s_0+b+1}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} \big ) . \end{aligned} \end{aligned}$$\end{document}

Moreover, if h is a quasi-periodic traveling wave, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}h$$\end{document} is a quasi-periodic traveling wave. In addition, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}h({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} ;

(iii) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(\,\cdot \,;\omega ), \,h(\,\cdot \,;\omega ) \in H_{0}^{s + 1}({\mathbb {T}}^{\nu + 2})$$\end{document} . Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q (v, h):= {{\mathcal {F}}}(v + h) - {{\mathcal {F}}}(v) - {{\mathcal {L}}} h = \lambda ^\theta {\mathcal {N}}(h,h) \end{aligned}$$\end{document}

satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {T}}^2} Q(v,h)({\varphi },x) \,\textrm{d}{x} = 0 , \quad \Vert Q(v, h) \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta } \Vert h \Vert _{s_0 + 1}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s + 1}^{\textrm{Lip}(\gamma )}\ . \end{aligned}$$\end{document}

Moreover, if v and h are quasi-periodic traveling waves, then Q(v, h) is a quasi-periodic traveling wave. In addition, assuming (3.2), if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v({\varphi },x),h({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(v,h)({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} .

Proof

The claimed estimates follow by (3.3), using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha -\theta <1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(v, h) = \lambda ^{\theta }{{\mathcal {N}}}(h, h)$$\end{document} , by a repeated applications of the interpolation estimate (2.2), and by the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{{\mathtt K}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{{\mathtt K}}^\perp $$\end{document} commute with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}$$\end{document} . Moreover, by (3.1) and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{div} {\mathfrak {B}}[h]= 0$$\end{document} for any sufficiently smooth function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h({\varphi }, x)$$\end{document} , by (1.8), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {T}}^2} Q(v,h)({\varphi },x) \,\textrm{d}{x} & = \lambda ^{\theta } \int _{{\mathbb {T}}^2} {\mathfrak {B}}[h({\varphi },x)] \cdot \nabla h({\varphi },x) \,\textrm{d}{x} \\ & = - \lambda ^\theta \int _{{\mathbb {T}}^2} \textrm{div} {\mathfrak {B}}[h({\varphi },x)] h({\varphi },x) \,\textrm{d}{x} = 0 . \end{aligned}$$\end{document}

The claims on the quasi-periodic traveling waves follow by (3.1), (3.3), the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi },x)$$\end{document} is a quasi-periodic function, and Lemmata 2.11, 2.12. Finally, assuming (3.2), the claims on the parities follow by (3.1), (1.5), (1.8), (3.3) and Lemma 2.9. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}_0$$\end{document} , we construct an approximate solution of (3.1) of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} v_{\textrm{app},N}({\varphi },x)&= v_0({\varphi },x)+v_1({\varphi },x)+...+v_N({\varphi },x) \\&= v_{\textrm{app},N-1}({\varphi },x) + v_{N}({\varphi },x), \quad v_{\textrm{app},-1}:= 0 , \end{aligned} \end{aligned}$$\end{document}

in such a way that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app}, N}({\varphi },x)$$\end{document} of size O(1) solves (3.1) up to a progressively smaller error term which, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}$$\end{document} sufficiently large, becomes perturbatively small for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} big enough, see (3.14) and (3.15) in Proposition 3.3. Moreover, both the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)$$\end{document} that we are going to construct and the final error term are quasi-periodic traveling waves.

For fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > \nu -1$$\end{document} , recalling (1.11), we define the set of the Diophantine frequency vectors

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt D}{\mathtt C}(\gamma ,\tau ):= \Big \{ \omega \in \Omega : \ |\omega \cdot \ell | \geqslant \gamma {\langle {\ell }\rangle }^{-\tau } \ \forall \,\ell \in {\mathbb {Z}}^{\nu }\setminus \{0\} \Big \} . \end{aligned}$$\end{document}

It is well known that the Lebesgue measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega {\setminus } {\mathtt D}{\mathtt C}(\gamma ,\tau ) | = O(\gamma )$$\end{document} . Moreover, we consider the following non-resonance condition, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > \nu + 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Omega _{\gamma }:= \Big \{ \omega&\in {\mathtt D}{\mathtt C}(\gamma ,\tau ) : | \lambda \,\omega \cdot \ell + \beta \,{\mathtt L}(j) | \geqslant \lambda \tfrac{\gamma }{ {\langle {\ell }\rangle }^{\tau }}, \forall \,(\ell ,j)\in {\mathbb {Z}}^{\nu +2}\setminus \{0\}, \ \pi ^\top (\ell ) + j=0 \Big \}. \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

At each step of the iteration, we solve linear equations of the following form.

Lemma 3.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > \nu + 1$$\end{document} , Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g (\,\cdot \,;\omega ,\lambda ) \in H_0^{s+2\tau +1}({\mathbb {T}}^{\nu +2})$$\end{document} be a quasi-periodic traveling wave. Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{\gamma }$$\end{document} as in (3.6), the linear equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \, \omega \cdot \partial _{{\varphi }} w({\varphi },x) + \beta \,{\mathtt L}w({\varphi },x) + g({\varphi },x) = 0 \end{aligned}$$\end{document}

is solved by the quasi-periodic traveling wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(\,\cdot ;\omega ,\lambda ) \in H_0^s({\mathbb {T}}^{\nu +2})$$\end{document} , defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} w({\varphi },x)&\, = w({\varphi },x;\omega ,\lambda ):= - (\lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L})^{-1} g({\varphi },x) \\&:= - \sum _{(\ell ,j)\in {\mathbb {Z}}^{\nu +2}, j\ne 0 \atop \pi ^\top (\ell )+j =0} \frac{1}{\textrm{i}\big ( \lambda \,\omega \cdot \ell + \beta \,{\mathtt L}(j)\big )} {\widehat{g}}(\ell ,j) e^{\textrm{i}(\ell \cdot {\varphi }+j\cdot x)}, \end{aligned} \end{aligned}$$\end{document}

with estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert w \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \lambda ^{-1} \gamma ^{-1} \Vert g \Vert _{s+2\tau +1}^{\textrm{Lip}(\gamma )} , \\&\quad \inf _{\omega \in \Omega _\gamma } \Vert w(\cdot ; \omega ) \Vert _s \geqslant \tfrac{1}{2} \min \{\lambda ^{-1}, |\beta |^{-1}\} \inf _{\omega \in \Omega _\gamma }\Vert g(\cdot ; \omega ) \Vert _{s-1} . \end{aligned} \end{aligned}$$\end{document}

Furthermore, we have the measure estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathtt D}{\mathtt C}(\gamma ,\tau ) {\setminus }\Omega _{\gamma }|\lesssim \gamma $$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(\gamma , \tau )$$\end{document} is defined in (3.5).

In addition, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} .

Proof

We write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w({\varphi },x) = \sum _{(\ell , j) \in {\mathbb {Z}}^{\nu +2}, \, j\ne 0} {{\widehat{w}}}(\ell ,j) e^{\textrm{i}(\ell \cdot {\varphi }+ j\cdot x)}$$\end{document} . Then, expanding the equation (3.7) in Fourier, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{i}\,\big ( \lambda \,\omega \cdot \ell + \beta \, {\mathtt L}(j) \big ) {{\widehat{w}}}(\ell ,j) + {{\widehat{g}}}(\ell ,j)= 0. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\varphi },x)$$\end{document} is a quasi-periodic traveling wave of the form (2.5), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\widehat{w}}}(\ell ,j):= {\left\{ \begin{array}{ll} \textrm{i}\,\big ( \lambda \,\omega \cdot \ell + \beta \, {\mathtt L}(j) \big )^{-1} {{\widehat{g}}}(\ell ,j) & \text {if } \ \pi ^\top (\ell ) + j = 0 , \\ 0 & \text {otherwise}. \end{array}\right. } \end{aligned}$$\end{document}

We conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w({\varphi },x)$$\end{document} in (3.8) is indeed a solution of (3.7) and it is a quasi-periodic traveling wave. The upper bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert w \Vert _{s}^{\textrm{Lip}(\gamma )}$$\end{document} in (3.9) follows by Definition 2.1 and since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{\gamma }$$\end{document} as in (3.6), whereas the lower bound follows by

the upper bound on the denominator, having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\omega |\leqslant 2$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{\gamma } \subset {\mathtt D}{\mathtt C}(\gamma ,\tau )$$\end{document} (see (3.5)-(3.6)),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\lambda \,\omega \cdot \ell +\beta {\mathtt L}(j) | \leqslant \lambda | \omega | |\ell | + |\beta | \leqslant 2 \max \{\lambda ,|\beta |\} |\ell | \leqslant 2 \max \{\lambda ,|\beta |\} {\langle {\ell ,j}\rangle }. \end{aligned}$$\end{document}

We now prove the measure estimate. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,j)\in {\mathbb {Z}}^{\nu +2}{\setminus }\{0\}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) +j=0$$\end{document} , we define the resonant set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} R_{\ell ,j}:= \Big \{ \omega \in {\mathtt D}{\mathtt C}(\gamma ,\tau ) \,: \, | \lambda \,\omega \cdot \ell + \beta \,{\mathtt L}(j) | < \lambda \gamma {\langle {\ell }\rangle }^{-\tau } \Big \} . \end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ne 0$$\end{document} . Indeed if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = 0$$\end{document} , then also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 0$$\end{document} (by the relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) +j=0$$\end{document} ) but this is not possible since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j ) \ne (0, 0)$$\end{document} . Hence, let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega = \frac{\ell }{|\ell |} s + v, \quad v \cdot \ell = 0 \end{aligned}$$\end{document}

and set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} z(s):=\lambda \,\omega \cdot \ell + \beta \,{\mathtt L}(j) = \lambda |\ell | s + \beta {\mathtt L}(j). \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\partial _s z(s)| \geqslant \lambda |\ell |$$\end{document} , one gets that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Big | \Big \{ s: \omega = \frac{\ell }{|\ell |} s + v \in R_{\ell , j} \Big \} \Big | \lesssim \lambda \gamma \langle \ell \rangle ^{- (\tau + 1)} \end{aligned}$$\end{document}

and hence by Fubini, one obtains that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|R_{\ell , j}| \lesssim \lambda \gamma \langle \ell \rangle ^{- (\tau + 1)}$$\end{document} . We then compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} | {\mathtt D}{\mathtt C}(\gamma ,\tau ) \setminus \Omega _{\gamma } |&= \Big | \bigcup _{(\ell ,j)\in {\mathbb {Z}}^{\nu +2}\setminus \{0\} \atop \pi ^\top (\ell )+j=0} R_{\ell ,j} \Big | = \Big | \bigcup _{{\begin{array}{c} \ell \in {\mathbb {Z}}^\nu \setminus \{0\} \\ |j| \lesssim |\ell | \end{array}}} R_{\ell ,j } \Big | \\&\lesssim \sum _{{\begin{array}{c} \ell \ne 0 \\ |j| \lesssim |\ell | \end{array}}} \frac{\lambda \gamma }{\langle \ell \rangle ^{\tau + 1}} \lesssim \sum _{\ell \ne 0} \frac{\lambda \gamma |\ell |^2}{\langle \ell \rangle ^{\tau + 1}} \lesssim \sum _{\ell \ne 0} \frac{\lambda \gamma }{\langle \ell \rangle ^{\tau - 1}} , \end{aligned} \end{aligned}$$\end{document}

where the last inequality holds since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > \nu + 1$$\end{document} .

Finally, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} , then we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \,\omega \cdot \partial _{{\varphi }}+\beta \,{\mathtt L}$$\end{document} is reversible, by (1.5), Lemma 2.9 and Definition 2.8. This concludes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We are now ready to construct explicitly the approximate solution of (3.1).

Proposition 3.3

(Approximate solution). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}_0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > \nu + 1$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{\gamma }$$\end{document} , there exists a quasi-periodic traveling wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x) = v_{\textrm{app},N}({\varphi },x; \omega ,\lambda ) \in H_0^s({\mathbb {T}}^{\nu +2})$$\end{document} of the form (3.4) satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {F}}(v_{\textrm{app},N}({\varphi },x)) = q_{N}({\varphi },x), \end{aligned}$$\end{document}

where the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N}({\varphi },x) = q_{N}({\varphi },x,\omega ,\lambda ) \in H_0^s({\mathbb {T}}^{\nu +2})$$\end{document} is a quasi-periodic traveling wave as well. Moreover, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\lambda }=\overline{\lambda }(s,N,\tau ,\beta )\gg 1$$\end{document} large enough such that, defining the constants

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\kappa _{N-\frac{1}{2}}(\tau ):= \kappa _{N}(\tau ) -1 , \quad \kappa _{N}(\tau ):= 2(N+1)(\tau +1) , \end{aligned} \end{aligned}$$\end{document}

and assuming, for a given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}\in (0,\frac{1}{3}(2-\alpha ))$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \geqslant \overline{\lambda } \geqslant \max \{ 1,|\beta |\}, \quad \gamma = \lambda ^{-{\mathtt c}} , \quad \theta := \alpha -1 +{\mathtt c}, \end{aligned}$$\end{document}

the following estimates hold, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert v_{\textrm{app},N} -v_{\textrm{app},N-1} \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _{s,N} \lambda ^{N(\alpha -2(1-{\mathtt c}) )} \Vert f \Vert _{s+\kappa _{N-\frac{1}{2}}(\tau )} \,; \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert v_{\textrm{app},N} \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _{s,N} \Vert f \Vert _{s+\kappa _{N-\frac{1}{2}}(\tau )}\,, \quad \inf _{\omega \in \Omega _\gamma } \Vert v_{\textrm{app},N}(\cdot ; \omega ) \Vert _s \gtrsim _{s, N} \lambda ^{-{\mathtt c}} \Vert f \Vert _{s-1} \,; \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert q_{N} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s,N} \lambda ^{(N+1)(\alpha -2(1-{\mathtt c}))+1-{\mathtt c}} \Vert f\Vert _{s+\kappa _{N}(\tau )}\,. \end{aligned}$$\end{document}

In particular, assuming the symmetry condition (3.2) on the forcing term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\varphi }, x)$$\end{document} in (1.9), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)= \textrm{odd}({\varphi },x)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_N({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} . Finally, suppose that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N >\frac{\alpha -(1-{\mathtt c})}{2(1-{\mathtt c})-\alpha }. \end{aligned}$$\end{document}

Then, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)$$\end{document} approximately solves (3.1), namely the term on the right-hand side of (3.14) becomes perturbatively small for sufficiently large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} .

Proof

We argue by induction on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}_0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},0}({\varphi },x) = v_0({\varphi },x)$$\end{document} be a solution of the equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\lambda \,\omega \cdot \partial _{{\varphi }} + \beta \, {\mathtt L}) v_{0}({\varphi },x) - \lambda ^{\alpha -\theta } f({\varphi },x) = 0. \end{aligned}$$\end{document}

By Lemma 3.2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_0({\varphi },x)$$\end{document} is a well-defined quasi-periodic traveling wave, with estimates, recalling (3.11),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert v_{0} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \lambda ^{-1} \gamma ^{-1} \Vert \lambda ^{\alpha -\theta } f \Vert _{s+2\tau +1} = \gamma ^{-1}\lambda ^{-{\mathtt c}} \Vert f\Vert _{s+2\tau +1} = \Vert f\Vert _{s+2\tau +1} , \\&\quad \inf _{\omega \in \Omega _\gamma }\Vert v_{0}(\cdot ; \omega ) \Vert _{s}\geqslant \frac{1}{2} \min \{\lambda ^{-1}, |\beta |^{-1} \} \, \Vert \lambda ^{\alpha -\theta } f \Vert _{s-1} \geqslant \frac{1}{2} \lambda ^{-{\mathtt c}} \Vert f \Vert _{s-1}, \end{aligned} \end{aligned}$$\end{document}

which proves (3.12) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=0$$\end{document} , as well as (3.13), having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{-\frac{1}{2}}(\tau )=2\tau +1$$\end{document} . Using (3.16), we have that (3.10) holds at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=0$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_0({\varphi },x)$$\end{document} defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_0({\varphi },x):= \lambda ^{\theta }{\mathcal {N}}(v_0({\varphi },x),v_0({\varphi },x)) . \end{aligned}$$\end{document}

By (3.1), (1.8), Lemma 2.2, Lemma 3.1-(iii), estimate (3.17) and (3.11), we obtain that (3.18) has zero average in space and satisfies (3.14) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=0$$\end{document} , having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{0}(\tau )= 2(\tau +1)>0$$\end{document} . The claim that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_0({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} follows by assuming (3.2) and by Lemma 3.2, whereas \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_0({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} follows by (3.18), (3.1) and Lemma 3.1-(iii).

We now assume by induction that (3.10) holds at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}_0$$\end{document} with estimates (3.12)-(3.13) and we prove the claim at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} . We insert (3.4) into (3.10), both with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\rightsquigarrow N+1$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} satisfies (3.10), we look for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{N+1}$$\end{document} satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\big (\lambda \, \omega \cdot \partial _{{\varphi }} + \beta \, {\mathtt L}\big )v_{N+1} + \lambda ^{\theta }\big ({\mathcal {N}}(v_{\textrm{app},N} + v_{N+1},v_{\textrm{app},N} + v_{N+1} ) \\&\quad - {\mathcal {N}}(v_{\textrm{app},N},v_{\textrm{app},N}) \big )+q_{N}({\varphi },x) = q_{N+1}({\varphi },x), \end{aligned} \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N+1}({\varphi },x)$$\end{document} to be determined. In particular, we choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{N+1}({\varphi },x)$$\end{document} to be a solution to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big (\lambda \, \omega \cdot \partial _{{\varphi }} + \beta \, {\mathtt L}\big )v_{N+1}({\varphi },x)+q_{N}({\varphi },x) =0. \end{aligned}$$\end{document}

Then, by Lemma 3.2, estimate (3.14) and (3.11), we obtain, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert v_{N+1} \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} \lambda ^{-1} \gamma ^{-1} \Vert q_{N} \Vert _{s+2\tau + 1}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s} \lambda ^{-1+{\mathtt c}} \lambda ^{(N+1)(\alpha -2(1-{\mathtt c})) +1-{\mathtt c}} \Vert f \Vert _{s+ \kappa _{N}(\tau )+2\tau +1}^{\textrm{Lip}(\gamma )} , \end{aligned} \end{aligned}$$\end{document}

which proves (3.12) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} , setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{N+1-\frac{1}{2}}(\tau ):= \kappa _{N}(\tau )+2\tau +1$$\end{document} .

We now prove (3.13) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} . By (3.4), estimate (3.12) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} , the induction assumption on (3.13), (3.11) and assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert v_{\textrm{app},N+1} \Vert _{s}^{\textrm{Lip}(\gamma )}&\leqslant \Vert v_{\textrm{app},N} \Vert _{s}^{\textrm{Lip}(\gamma )} + \Vert v_{\textrm{app},N+1} - v_{\textrm{app},N} \Vert _{s}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s,N} \big ( 1 + \lambda ^{(N+1)(\alpha -2(1-{\mathtt c}))} \big ) \Vert f \Vert _{s+\kappa _{N-\frac{1}{2}}(\tau )}^{\textrm{Lip}(\gamma )} \lesssim _{s,N} \Vert f \Vert _{s+\kappa _{N-\frac{1}{2}}(\tau )}^{\textrm{Lip}(\gamma )}, \end{aligned} \end{aligned}$$\end{document}

which proved the upper bound in (3.13) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} , whereas

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \inf _{\omega \in \Omega _\gamma }\Vert v_{\textrm{app},N+1}(\cdot ; \omega ) \Vert _{s}&\geqslant \inf _{\omega \in \Omega _\gamma } \Vert v_{\textrm{app},N}(\cdot ; \omega ) \Vert _{s} - \Vert v_{\textrm{app},N+1} - v_{\textrm{app},N}\Vert _{s}^{\textrm{Lip}(\gamma )} \\&\geqslant C_{s, N}\lambda ^{-{\mathtt c}} \Vert f \Vert _{s-1} - K_{s, N} \lambda ^{(N+1)(\alpha -2(1-{\mathtt c}))} \Vert f\Vert _{s+\kappa _{N+1-\frac{1}{2}}} \\&\geqslant \frac{C_{s, N}}{2} \lambda ^{-{\mathtt c}} \Vert f \Vert _{s-1} , \end{aligned} \end{aligned}$$\end{document}

(for some positive constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{s, N}, K_{s, N} > 0$$\end{document} ), assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \geqslant {\overline{\lambda }} \gg 1$$\end{document} large enough such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda ^{(N+1)(\alpha -2(1-{\mathtt c})) + {\mathtt c}}\leqslant \frac{C_{s,N}}{2 K_{s,N}}\frac{ \Vert f \Vert _{s-1} }{ \Vert f\Vert _{s+\kappa _{N+1-\frac{1}{2}}} }, \end{aligned}$$\end{document}

which shows the lower bound in (3.13) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} , as claimed. Note that we can impose (3.22) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}<\frac{1}{3}(2-\alpha )$$\end{document} implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(1-{\mathtt c})-\alpha> {\mathtt c}> \frac{{\mathtt c}}{N+1}$$\end{document} .

Furthermore, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} are quasi-periodic traveling waves by induction assumption, by Lemma 3.2 we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{N+1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N+1}$$\end{document} are quasi-periodic traveling waves as well.

It remains to define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N+1}({\varphi },x)$$\end{document} and to prove (3.14) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} . By inserting (3.20) into (3.19), the identity holds by defining

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} q_{N+1}&:=\lambda ^{\theta }\big ({\mathcal {N}}(v_{\textrm{app},N} + v_{N+1},v_{\textrm{app},N} + v_{N+1} ) - {\mathcal {N}}(v_{\textrm{app},N},v_{\textrm{app},N}) \big ) \\&\, = \lambda ^{\alpha -1+{\mathtt c}} \big ( {\mathcal {N}}(v_{\textrm{app},N}, v_{N+1} ) + {\mathcal {N}}( v_{N+1},v_{\textrm{app},N} ) + {\mathcal {N}}( v_{N+1}, v_{N+1} ) \big ), \end{aligned} \end{aligned}$$\end{document}

where we used (3.11) and the bilinearity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}(\,\cdot \,, \,\cdot \,)$$\end{document} . By Lemma 3.1-(iii), we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N+1}({\varphi },x)$$\end{document} has zero average in space. By (3.1), (1.8), Lemma 2.2, and estimates (3.13), (3.21), we estimate (3.23) by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert q_{N+1} \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} \lambda ^{\alpha -1+{\mathtt c}} \Big ( \Vert v_{\textrm{app},N} \Vert _{s-1}^{\textrm{Lip}(\gamma )} \Vert v_{N+1} \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} + \Vert v_{\textrm{app},N} \Vert _{s_0-1}^{\textrm{Lip}(\gamma )} \Vert v_{N+1} \Vert _{s+1}^{\textrm{Lip}(\gamma )} \\&\qquad \qquad \quad \ \ \Vert v_{N+1} \Vert _{s-1}^{\textrm{Lip}(\gamma )} \Vert v_{\textrm{app},N} \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} + \Vert v_{N+1} \Vert _{s_0-1}^{\textrm{Lip}(\gamma )} \Vert v_{\textrm{app},N} \Vert _{s+1}^{\textrm{Lip}(\gamma )} \\&\qquad \qquad \quad \ \ \Vert v_{N+1} \Vert _{s-1}^{\textrm{Lip}(\gamma )} \Vert v_{N+1} \Vert _{s_0+1}^{\textrm{Lip}(\gamma )} + \Vert v_{N+1} \Vert _{s_0-1}^{\textrm{Lip}(\gamma )} \Vert v_{N+1} \Vert _{s+1}^{\textrm{Lip}(\gamma )} \Big ) \\&\lesssim _{s,N} \lambda ^{(N+2)(\alpha -2(1-{\mathtt c})) + 1-{\mathtt c}} \big ( 1+ \lambda ^{(N+1)(\alpha -2(1-{\mathtt c}))} \big ) \Vert f \Vert _{s+\kappa _{N+1-\frac{1}{2}}(\tau )+1} \\&\lesssim _{s,N} \lambda ^{(N+2)(\alpha -2(1-{\mathtt c}))+1-{\mathtt c}} \Vert f \Vert _{s+\kappa _{N+1-\frac{1}{2}}(\tau )+1} , \end{aligned} \end{aligned}$$\end{document}

which yields (3.14) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N+1$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{N+1}(\tau ):= \kappa _{N+1-\frac{1}{2}}(\tau )+1$$\end{document} . Furthermore, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} is a quasi-periodic traveling wave by induction assumption and we showed that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{N+1}$$\end{document} is a quasi-periodic traveling wave, it follows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N+1}$$\end{document} in (3.23) is a quasi-periodic traveling wave as well. Finally, the claim that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N+1}({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} follows from the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_N({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} by induction assumption and by Lemma 3.2, whereas the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{N+1}({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} follows by (3.23), (3.1), (1.8) and Lemma 2.9. This concludes the proof of the claim and of the proposition. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The Linearized Operator

Once the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}({\varphi },x)$$\end{document} with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} (as constructed in Proposition 3.3, see (3.13) and (3.14)), is obtained, we proceed to study the linearization of equation (3.1) at any approximate solution of the Nash-Moser iteration (see Sect. 7). This approximate solution takes the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w= v_{\textrm{app},N}+ g \in {{\mathcal {C}}}^\infty ({\mathbb {T}}^\nu \times {\mathbb {T}}^2) , \end{aligned}$$\end{document}

satisfying the ansatz

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w \Vert _{s_0 + \sigma }^{\textrm{Lip}(\gamma )} =\Vert w \Vert _{s_0 + \sigma ,\Lambda _{o}}^{\textrm{Lip}(\gamma )} \leqslant C_0, \quad \begin{array}{ll} \text {for some constants } \sigma ,\, C_0 \gg 0 \text { large enough } \\ \text {and for a given } \Lambda _{o}\subseteq \Omega _{\gamma } \subset {\mathbb {R}}^\nu , \end{array} \end{aligned}$$\end{document}

recalling the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\gamma }$$\end{document} in (3.6). The set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{o}=\Lambda _{o}(w)$$\end{document} will change at each step of the Nash-Moser iteration in Proposition 7.3. Recalling (3.3), the linearized operator is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}:= \textrm{d}_{v} {\mathcal {F}}( w):= \lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ \lambda ^{\theta }\textbf{a}_{0}\cdot \nabla + \lambda ^{\theta } {{\mathcal {E}}_{0}}, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{a}_{0}:= {\mathfrak {B}}\big [ w \big ] , \quad {\mathcal {E}}_{0}[h]:= \nabla w \cdot {\mathfrak {B}}[h] \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta := \alpha -1+{\mathtt c}, \quad \text {for some} \ \ {\mathtt c}\in (0,\tfrac{1}{3}(2-\alpha )). \end{aligned}$$\end{document}

The range for the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}>0$$\end{document} ensures that, for any fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta -1< \theta - 1 + {\mathtt c}< 0 . \end{aligned}$$\end{document}

The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt c}$$\end{document} will be fixed arbitrarily small enough in Proposition 7.3, independently of the step of the Nash-Moser iteration. In particular, it will guarantee that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda ^{\theta -1} \gamma ^{-1} \ll 1 . \end{aligned}$$\end{document}

By the ansatz (4.2) and by the definitions (4.4), we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \textbf{a}_{0} \Vert _{s} \lesssim _s \Vert w \Vert _{s-1} , \quad \forall s \geqslant s_0 , \end{aligned}$$\end{document}

and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{0} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{- 1}$$\end{document} , with the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{\mathcal {E}}_{0}|_{- 1, s, \alpha } \lesssim _{s, \alpha } \Vert w \Vert _{s+1} , \quad \forall s \geqslant s_0, \quad \alpha \in {\mathbb {N}}. \end{aligned}$$\end{document}

By the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{a}_{0}$$\end{document} in (4.4) and the definition of the Biot-Savart operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {B}}$$\end{document} in (1.8), we clearly have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\langle {\textbf{a}_{0}}\rangle }_{x}:=\frac{1}{(2\pi )^2}\int _{{\mathbb {T}}^2}\textbf{a}_{0}({\varphi },x) \,\textrm{d}{x} = 0, \quad \textrm{div}(\textbf{a}_{0}):= \nabla \cdot \textbf{a}_{0} = 0 . \end{aligned}$$\end{document}

Moreover, using also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{div} (\nabla ^\perp h) = 0$$\end{document} for any h, the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{a}_{0} \cdot \nabla $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal L}$$\end{document} leave invariant the subspace of zero average function, with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&[\Pi _0^\bot , \textbf{a}_{0} \cdot \nabla ] = 0, \quad [\Pi _0^\bot , {\mathcal {E}}_{0} ] = 0, \\&\quad \textbf{a}_{0} \cdot \nabla \Pi _0 = \Pi _0 \textbf{a}_{0} \cdot \nabla = 0, \quad {\mathcal {E}}_{0} \Pi _0 = \Pi _0 {\mathcal {E}}_{0} = 0 ,\\&\quad [\Pi _0^\bot , {{\mathcal {L}}}] = 0, \quad \Pi _0 {{\mathcal {L}}}= {{\mathcal {L}}} \Pi _0 = 0, \end{aligned} \end{aligned}$$\end{document}

implying that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{a}_{0} \cdot \nabla = \Pi _0^\bot \textbf{a}_{0} \cdot \nabla \Pi _0^\bot , \quad {\mathcal {E}}_{0} = \Pi _0^\bot {\mathcal {E}}_{0} \Pi _0^\bot , \quad {\mathcal L} = \Pi _0^\bot {{\mathcal {L}}} \Pi _0^\bot . \end{aligned}$$\end{document}

We always work on the space of zero average functions and we shall preserve this invariance along the whole paper.

Normal Form Reduction

We now present the step-by-step scheme of reduction for the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}$$\end{document} defined in (4.3).

Reduction of the transport

We consider the composition operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {B}}u)({\varphi },x):= h ({\varphi }, x+ {\varvec{\beta }}({\varphi },x)) , \quad ({\mathcal {B}}^{-1} h ) ({\varphi },y):= h ({\varphi }, y + \breve{{\varvec{\beta }}}({\varphi },y)), \end{aligned}$$\end{document}

induced by a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi }$$\end{document} -dependent family of diffeomorphisms of the torus

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathbb {T}}^2 \rightarrow {\mathbb {T}}^2, \quad x \mapsto y:= x + {\varvec{\beta }}({\varphi },x) , \end{aligned}$$\end{document}

with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \mapsto x = y + \breve{{\varvec{\beta }}}({\varphi },y)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\varvec{\beta }}\Vert _{s}$$\end{document} sufficiently small and where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}({\varphi },y)$$\end{document} are quasi-periodic traveling wave functions.

We need to reduce to constant coefficients the highest order operator which is a transport operator of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {T}}&:= \lambda \, \omega \cdot \partial _{{\varphi }} + \lambda ^{\theta } \textbf{a}_{0} \cdot \nabla = \lambda \, {\widetilde{{\mathcal {T}}}}, \\ {\widetilde{{\mathcal {T}}}}&:= \omega \cdot \partial _\varphi + \textbf{b}(\varphi , x)\cdot \nabla , \quad \textbf{b}(\varphi , x):= \varepsilon \textbf{a}_0(\varphi , x), \quad \varepsilon := \lambda ^{\theta -1}. \end{aligned} \end{aligned}$$\end{document}

Note that, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , by(4.5)-(4.6), one has that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon = \lambda ^{\theta -1} \ll 1$$\end{document} and the quasi-periodic traveling wave function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}(\varphi , x)$$\end{document} satisfies the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \textbf{b} \Vert _s \lesssim _s \lambda ^{\theta -1} \Vert w \Vert _{s-1}, \quad \forall s \geqslant s_0, \\&\quad \Vert \Delta _{12} \textbf{b} \Vert _{s_1} \lesssim _{s_1} \lambda ^{\theta -1} \Vert w_1 - w_2 \Vert _{s_1-1} , \quad s_1 \geqslant s_0, \end{aligned} \end{aligned}$$\end{document}

where w is given in (4.1), the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}$$\end{document} is given in (2.2), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1, w_2$$\end{document} satisfy the ansatz (4.2). We consider the set of Diophantine non-resonance conditions as defined in (3.5).

We have the following result.

Proposition 5.1

(Straightening of the transport operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document} ). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >\nu -1$$\end{document} . There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma := \sigma (\tau , \nu ) > 0$$\end{document} large enough such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma $$\end{document} , there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau , \nu ) \in (0, 1)$$\end{document} small enough and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}=\tau _{1}(\tau ,\nu )>0$$\end{document} such that, if (4.2), (5.3) hold and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_0^{\tau _{1}}\varepsilon \gamma ^{- 1} = N_0^{\tau _{1}} \lambda ^{\theta -1} \gamma ^{-1} \leqslant \delta , \end{aligned}$$\end{document}

is fulfilled, then the following holds. There exists an invertible diffeomorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2 \rightarrow {\mathbb {T}}^2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto x + {\varvec{\beta }}(\varphi , x; \omega )$$\end{document} with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \mapsto y + \breve{\varvec{\beta }}(\varphi , y; \omega )$$\end{document} , defined for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma , \tau )\cap \Lambda _{o}$$\end{document} , with the set given in (3.5),

satisfying, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s \leqslant S - \sigma $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\varvec{\beta }}\Vert _s^{\textrm{Lip}(\gamma )}, \Vert \breve{\varvec{\beta }}\Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_0^{2\tau +1} \varepsilon \gamma ^{- 1} \Vert w \Vert _{s + \sigma }^{\textrm{Lip}(\gamma )} , \end{aligned}$$\end{document}

such that one gets the conjugation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {B}}}^{- 1} {{\mathcal {T}}} {{\mathcal {B}}} = \lambda \, \omega \cdot \partial _\varphi , \end{aligned}$$\end{document}

where the invertible maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}^{-1}$$\end{document} are defined in (5.1), satisfying the estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {{\mathcal {B}}}^{\pm 1} h\Vert _s^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert h \Vert _s^{\textrm{Lip}(\gamma )} + \Vert w \Vert _{s + \sigma }^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0}^{\textrm{Lip}(\gamma )}, \\&\quad \Vert ({{\mathcal {B}}}^{\pm 1} - \textrm{Id}) h\Vert _s^{\textrm{Lip}(\gamma )} \lesssim _{s} N_0^{2\tau +1} \varepsilon \gamma ^{- 1} \big ( \Vert w \Vert _{s_0 + \sigma }^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s + 1}^{\textrm{Lip}(\gamma )} + \Vert w \Vert _{s + \sigma }^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0 + 1}^{\textrm{Lip}(\gamma )} \big ) . \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 \geqslant s_0$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1, w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma _{0} \geqslant s_1 + \sigma $$\end{document} . Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma , \tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1,w_2$$\end{document} satisfying (4.2) and recalling the notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}$$\end{document} in (2.2), one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \Delta _{12} {\varvec{\beta }}\Vert _{s_1} ,\, \Vert \Delta _{12} \breve{\varvec{\beta }}\Vert _{s_1} \lesssim _{s_1} N_0^{2\tau +1}\varepsilon \gamma ^{- 1} \Vert w_1 - w_2 \Vert _{s_1 + \sigma }, \\&\quad \Vert \Delta _{12} {{\mathcal {B}}}^{\pm 1} h \Vert _{s_1} \lesssim _{s_1} N_0^{2\tau +1} \varepsilon \gamma ^{- 1} \Vert w_1 -w_2 \Vert _{s_1 +\sigma } \Vert h \Vert _{s_1 + 1}. \end{aligned} \end{aligned}$$\end{document}

Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }},\breve{{\varvec{\beta }}}$$\end{document} are quasi-periodic traveling waves, and the related maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}, {{\mathcal {B}}}^{- 1}$$\end{document} are momentum preserving (see Definition 2.10). In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)= \textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }},\breve{{\varvec{\beta }}}$$\end{document} are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}(\varphi ,x)$$\end{document} and the related maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}, {{\mathcal {B}}}^{- 1}$$\end{document} are reversibility preserving (see Definition 2.8).

In order to prove Proposition 5.1, we follow [7] and [9]. First we show the following iterative lemma. We fix the constants

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&N_0 >0 , \quad N_{-1}:= 1, \quad \chi := 3/2, \quad N_{n}:= N_0^{\chi ^n}, \quad n \in {\mathbb {N}}_0, \\&\quad {\mathfrak {a}}:= 6 \tau +8, \quad {\mathfrak {b}}:= {\mathfrak {a}}+1 . \end{aligned} \end{aligned}$$\end{document}

Lemma 5.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >\nu -1$$\end{document} . There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma := \sigma (\tau , \nu ) > 0$$\end{document} large enough such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma $$\end{document} , there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau , \nu ) \in (0, 1)$$\end{document} small enough, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_0=N_0(S,\tau ,\nu )>0$$\end{document} large enough and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}=\tau _{1}(\tau ,\nu )>0$$\end{document} such that, if (4.2), (5.3) and (5.4) hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{0}\geqslant s_0 +\sigma $$\end{document} , the following holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} :

There exists a linear operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {T}}_{n}:= \lambda \,{\widetilde{{\mathcal {T}}}}_{n}, \quad {\widetilde{{\mathcal {T}}}}_{n}:= \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n} \cdot \nabla + \textbf{b}_{n}({\varphi },x) \cdot \nabla , \end{aligned} \end{aligned}$$\end{document}

defined for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n}^{\gamma }$$\end{document} , where we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}_{0}^{\gamma }:=\Lambda _{o} $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} and, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{\mathcal {O}}_{n}^{\gamma }:= \Big \{ \omega \in {\mathcal {O}}_{n-1}^\gamma \,: \, |\omega \cdot \ell + {\mathtt m}_{n-1}\cdot j | \geqslant {\gamma }\,{{\langle {\ell }\rangle }^{-\tau }} , \ \ \forall \,(\ell ,j)\in {\mathbb {Z}}^{\nu +2}, \\&\quad \ \ 0<| \ell | \leqslant N_{n-1} , \ \ \pi ^\top (\ell ) + j = 0 \Big \} , \end{aligned} \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt m}_{n}$$\end{document} satisfy the following properties. The function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}({\varphi },x)$$\end{document} is a quasi-periodic traveling wave, with estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \textbf{b}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} \leqslant C(s,{\mathfrak {b}}) N_{n-1}^{-{\mathfrak {a}}} \Vert \textbf{b}\Vert _{s+\sigma } , \quad \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} \leqslant C(s,{\mathfrak {b}}) N_{n-1} \Vert \textbf{b}\Vert _{s+\sigma }, \end{aligned}$$\end{document}

for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(s,{\mathfrak {b}})>0$$\end{document} monotone increasing with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[s_0,S-\sigma ]$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt m}_{n}$$\end{document} is a real constant satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | {\mathtt m}_{n} |^{\textrm{Lip}(\gamma )} \leqslant 2 \Vert \textbf{b}\Vert _{s_0}^{\textrm{Lip}(\gamma )} , \quad |{\mathtt m}_{n} - {\mathtt m}_{n-1} |^{\textrm{Lip}(\gamma )} \leqslant C(s_0,{\mathfrak {b}}) N_{n-2}^{-{\mathfrak {a}}} \Vert \textbf{b}\Vert _{s_0+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} , \ \ n\geqslant 2. \nonumber \\ \end{aligned}$$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 1$$\end{document} , there exists an invertible diffeomorphism of the torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2\rightarrow {\mathbb {T}}^2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto x + {\varvec{\beta }}_{n-1}({\varphi },x)$$\end{document} , with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \mapsto y + \breve{{\varvec{\beta }}}_{n-1}({\varphi },x)$$\end{document} , such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [s_0,S - \sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {\varvec{\beta }}_{n-1} \Vert _{s}^{\textrm{Lip}(\gamma )}, \ \Vert \breve{{\varvec{\beta }}}_{n-1} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_{n-1}^{2\tau +1} N_{n-1}^{-{\mathfrak {a}}} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \\&\quad \Vert {\varvec{\beta }}_{n-1} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} , \ \Vert \breve{{\varvec{\beta }}}_{n-1} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_{n-1}^{2\tau +1} N_{n-1} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

The operators

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {B}}_{n-1} u)({\varphi },x):= h ({\varphi }, x+ {\varvec{\beta }}_{n-1}({\varphi },x)) , \quad ({\mathcal {B}}_{n-1}^{-1} h ) ({\varphi },y):= h ({\varphi }, y + \breve{{\varvec{\beta }}}_{n-1}({\varphi },y)), \end{aligned}$$\end{document}

satisfy, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n}^{\gamma }$$\end{document} , the conjugation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {T}}_{n} = {\mathcal {B}}_{n-1}^{-1} {\mathcal {T}}_{n-1} {\mathcal {B}}_{n-1} . \end{aligned}$$\end{document}

Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}_{n-1},\breve{{\varvec{\beta }}}_{n-1}$$\end{document} are quasi-periodic traveling waves, and the related maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_{n-1}, {\mathcal B}_{n-1}^{- 1}$$\end{document} are momentum preserving. In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}_{n-1},\breve{{\varvec{\beta }}}_{n-1}$$\end{document} are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}(\varphi ,x)$$\end{document} , the related maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {B}}}_{n-1}, {\mathcal B}_{n-1}^{- 1}$$\end{document} are reversibility preserving, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi },x)$$\end{document} .

Proof

We argue by induction on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} . The claimed statements for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} follow directly by setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt m}_{0}:= 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{0}({\varphi },x):=\textbf{b}({\varphi },x)$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}({\varphi },x)$$\end{document} defined in (5.2) with estimate (5.3).

We now assume that the claimed properties hold for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} and we prove them at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . We look for a diffeomorphism of the torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}^2 \rightarrow {\mathbb {T}}^2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \mapsto x + {\varvec{\beta }}_{n}({\varphi },x)$$\end{document} , with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \mapsto y + \breve{{\varvec{\beta }}}_{n}({\varphi },x)$$\end{document} , such that, defining the operators

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {B}}_{n} u)({\varphi },x):= h ({\varphi }, x+ {\varvec{\beta }}_{n}({\varphi },x)) , \quad ({\mathcal {B}}_{n}^{-1} h ) ({\varphi },y):= h ({\varphi }, y + \breve{{\varvec{\beta }}}_{n}({\varphi },y)), \end{aligned}$$\end{document}

the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{n}^{-1} {\mathcal {T}}_{n} {\mathcal {B}}_{n}= \lambda \, {\mathcal {B}}_{n}^{-1} {\widetilde{{\mathcal {T}}}}_{n} {\mathcal {B}}_{n}$$\end{document} has the desired properties. We compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {B}}_{n}^{-1} {\widetilde{{\mathcal {T}}}}_{n} {\mathcal {B}}_{n}&= \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n} \cdot \nabla + \big \{ {\mathcal {B}}_{n}^{-1} \big ( \omega \cdot \partial _{{\varphi }} {\varvec{\beta }}_{n} + {\mathtt m}_{n} \cdot \nabla {\varvec{\beta }}_{n} + \textbf{b}_{n} + \textbf{b}_{n} \cdot \nabla {\varvec{\beta }}_{n} \big ) \big \} \cdot \nabla \\&= \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n} \cdot \nabla + \big \{ {\mathcal {B}}_{n}^{-1} \big ( \omega \cdot \partial _{{\varphi }} {\varvec{\beta }}_{n} + {\mathtt m}_{n} \cdot \nabla {\varvec{\beta }}_{n} + \Pi _{N_n} \textbf{b}_{n} \big ) \big \} \cdot \nabla + \textbf{b}_{n+1} \cdot \nabla , \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{b}_{n+1}:= {\mathcal {B}}_{n}^{-1} \textbf{g}_{n} , \quad \textbf{g}_{n}:= \Pi _{N_n}^\perp \textbf{b}_{n} + \textbf{b}_{n} \cdot \nabla {\varvec{\beta }}_{n} , \end{aligned}$$\end{document}

and the projectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{N_n}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{N_n}^\perp $$\end{document} are defined in (2.6). For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n+1}^\gamma $$\end{document} , we solve the homological equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega \cdot \partial _{{\varphi }} {\varvec{\beta }}_{n} + {\mathtt m}_{n} \cdot \nabla {\varvec{\beta }}_{n} + \Pi _{N_n} \textbf{b}_{n} = {\langle {\textbf{b}_{n}}\rangle }_{{\varphi }} , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\langle {\textbf{b}_{n}}\rangle }_{{\varphi }}:= \frac{1}{(2\pi )^{\nu }} \int _{{\mathbb {T}}^{\nu }} \textbf{b}_{n}({\varphi },x) \,\textrm{d}{{\varphi }}. \end{aligned}$$\end{document}

Note that, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}$$\end{document} is a quasi-periodic traveling wave, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle {\textbf{b}_{n}}\rangle }_{{\varphi }}={\langle {\textbf{b}_{n}}\rangle }_{{\varphi },x}\in {\mathbb {R}}^2$$\end{document} is constant, since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{(2\pi )^\nu } \int _{{\mathbb {T}}^\nu }\textbf{b}_{n}({\varphi },x)\,\textrm{d}{{\varphi }}= \sum _{j\in {\mathbb {Z}}^2 \atop \pi ^\top (0)+j=0} {\widehat{\textbf{b}_{n}}}(0,j) e^{\textrm{i}\,jx} ={\widehat{\textbf{b}_{n}}}(0,0) ={\langle {\textbf{b}_{n}}\rangle }_{{\varphi },x}. \end{aligned}$$\end{document}

Explicitly, the solution of the equation (5.17) is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\varvec{\beta }}_{n}({\varphi },x)&:= -( \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n} \cdot \nabla )^{-1} \big [ \Pi _{N_n}\textbf{b}_{n} - {\langle {\textbf{b}_{n}}\rangle }_{{\varphi }} \big ] \\&:= - \sum _{(\ell , j) \in {\mathbb {Z}}^{\nu +2} \setminus \{0\}, \atop |\ell |\leqslant N_n, \, \pi ^\top (\ell ) + j = 0} \frac{1}{\textrm{i}(\omega \cdot \ell + {\mathtt m}_{n}\cdot j)} {\widehat{\textbf{b}}}_{n}(\ell ,j) e^{\textrm{i}(\ell \cdot {\varphi }+ j\cdot x)} . \end{aligned} \end{aligned}$$\end{document}

We define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {T}}_{n+1}&:= \lambda \, {\widetilde{{\mathcal {T}}}}_{n+1}, \quad {\widetilde{{\mathcal {T}}}}_{n+1}:= \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n+1} \cdot \nabla + \textbf{b}_{n+1} \cdot \nabla , \\ {\mathtt m}_{n+1}&:= {\mathtt m}_{n} + {\langle {\textbf{b}_{n}}\rangle }_{{\varphi }} . \end{aligned} \end{aligned}$$\end{document}

Therefore, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n+1}^\gamma $$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathcal {T}}}}_{n+1}= {\mathcal {B}}_{n}^{-1} {\widetilde{{\mathcal {T}}}}_{n} {\mathcal {B}}_{n}$$\end{document} , so that (5.15) holds at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . By (5.18) and Lemma 2.4, we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert {\varvec{\beta }}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} \Vert \Pi _{N_n} \textbf{b}_{n} \Vert _{s+2\tau +1}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_n^{2\tau +1} \gamma ^{-1} \Vert \textbf{b}_{n}\Vert _{s}^{\textrm{Lip}(\gamma )} , \\ \Vert \nabla {\varvec{\beta }}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} \Vert \Pi _{N_n} \textbf{b}_{n} \Vert _{s+2\tau +2}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_n^{2\tau +2} \gamma ^{-1} \Vert \textbf{b}_{n}\Vert _{s}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

The latter estimate, together with the induction estimate (5.12) on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}$$\end{document} , imply, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert {\varvec{\beta }}_{n+1} \Vert _{s}^{\textrm{Lip}(\gamma )} , \Vert \breve{{\varvec{\beta }}}_{n+1} \Vert _{s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} N_n^{2\tau +1} N_{n-1}^{-{\mathfrak {a}}} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \\ \Vert {\varvec{\beta }}_{n+1} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} , \Vert \breve{{\varvec{\beta }}}_{n+1} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )}&\lesssim _{s} N_n^{2\tau +1} N_{n-1} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \end{aligned} \end{aligned}$$\end{document}

which are the estimates (5.14) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . Note that, by the definition of the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {a}}$$\end{document} in (5.9) and the ansatz (4.2), from (5.21) we deduce, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=s_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\varvec{\beta }}_{n+1}\Vert _{s_0}^{\textrm{Lip}(\gamma )} , \Vert \breve{{\varvec{\beta }}}_{n+1}\Vert _{s_0}^{\textrm{Lip}(\gamma )} \lesssim N_0^{2\tau +1} \varepsilon \gamma ^{-1}. \end{aligned}$$\end{document}

Together with the smallness condition (5.4), Lemma 2.14 implies that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S-\sigma +{\mathfrak {b}}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\mathcal {B}}_{n}^{\pm 1} h\Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert h \Vert _{s}^{\textrm{Lip}(\gamma )} + N_n^{2\tau +1} \gamma ^{-1} \Vert \textbf{b}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0}^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Now we estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n+1}$$\end{document} in (5.16). First, we estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{g}_{n}$$\end{document} . By (5.9), (5.12), the ansatz (4.2) and the smallness condition (5.4), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_n^{2\tau +2} \gamma ^{-1} \Vert \textbf{b}_{n} \Vert _{s_0}^{\textrm{Lip}(\gamma )} \leqslant 1. \end{aligned}$$\end{document}

By Lemma 2.4, Lemma 2.2 and estimates (5.20), (5.23), we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \textbf{g}_{n} \Vert _{s_0}^{\textrm{Lip}(\gamma )} \lesssim \Vert \textbf{b}_{n}\Vert _{s_0}^{\textrm{Lip}(\gamma )} , \\&\quad \Vert \textbf{g}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_n^{-{\mathfrak {b}}} \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} + N_{n-1}^{2\tau +2} \gamma ^{-1} \Vert \textbf{b}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} \Vert \textbf{b}_{n} \Vert _{s_0}^{\textrm{Lip}(\gamma )} , \\&\quad \Vert \textbf{g}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} (1 + N_{n}^{2\tau +2} \gamma ^{-1} \Vert \textbf{b}_{n} \Vert _{s_0}^{\textrm{Lip}(\gamma )}) \lesssim _{s} \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

Therefore, estimates (5.22)-(5.24) imply that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \textbf{b}_{n+1} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_n^{-{\mathfrak {b}}} \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} + N_n^{2\tau +2} \gamma ^{-1} \Vert \textbf{b}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} \Vert \textbf{b}_{n} \Vert _{s_0}^{\textrm{Lip}(\gamma )} , \\&\quad \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} \lesssim _{s} \Vert \textbf{b}_{n} \Vert _{s+{\mathfrak {b}}}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

Using the definition of the constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {a}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {b}}$$\end{document} in (5.9) and the induction assuptions on the estimates (5.12), by a standard argument we deduce the estimates (5.12) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . The estimates (5.13) follow by the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt m}_{n+1}$$\end{document} in (5.19), the induction assumption on (5.12) and by a telescopic argument.

Finally, by (5.18), since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}({\varphi },x)$$\end{document} is a quasi-periodic traveling wave by induction assumption, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}_{n}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}_{n}({\varphi },y)$$\end{document} are quasi-periodic traveling waves by Lemma 2.15, which also imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{n}^{\pm 1}$$\end{document} are momentum preserving operators. Recalling (5.16), we deduce that also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n+1}({\varphi },y)$$\end{document} is a quasi-periodic traveling wave.

Finally if w is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi }, x)$$\end{document} , by (5.18), since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n}({\varphi },x)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi },x)$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}_{n}({\varphi },x)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}_{n}({\varphi },y)$$\end{document} are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi },y)$$\end{document} and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{n}^{\pm 1}$$\end{document} are reversibility preseving operators. Recalling (5.16), we deduce that also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{n+1}({\varphi },y)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi },y)$$\end{document} . This concludes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We then define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widetilde{{\mathcal {B}}}}_{n}:= {\mathcal {B}}_{0} \circ {\mathcal {B}}_{1} \circ ... \circ {\mathcal {B}}_{n} , \quad \text {with inverse} \quad {\widetilde{{\mathcal {B}}}}_{n}^{-1} = {\mathcal {B}}_{n}^{-1} \circ ... \circ {\mathcal {B}}_{1}^{-1} \circ {\mathcal {B}}_{0}^{-1} . \end{aligned}$$\end{document}

Lemma 5.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >\nu -1$$\end{document} . There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma := \sigma (\tau , \nu ) > 0$$\end{document} large enough such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma $$\end{document} , there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau , \nu ) \in (0, 1)$$\end{document} small enough, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_0=N_0(S,\tau ,\nu )>0$$\end{document} large enough and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}=\tau _{1}(\tau ,\nu )>0$$\end{document} such that, if (4.2), (5.3) and (5.4) hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{0}\geqslant s_0 +\sigma $$\end{document} , the following hold:

(i) For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in {\mathbb {N}}_0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\widetilde{{\mathcal {B}}}}_{n} u)({\varphi },x):= h ({\varphi }, x+ {\varvec{\alpha }}_{n}({\varphi },x)) , \quad ({\widetilde{{\mathcal {B}}}}_{n}^{-1} h ) ({\varphi },y):= h ({\varphi }, y + \breve{{\varvec{\alpha }}}_{n}({\varphi },y)), \end{aligned}$$\end{document}

for some functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}_{n}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\alpha }}}_{n}({\varphi },y)$$\end{document} such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {\varvec{\alpha }}_{0}\Vert _{s}^{\textrm{Lip}(\gamma )}, \ \Vert \breve{{\varvec{\alpha }}}_{0}\Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_0^{2\tau +1} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \\&\quad \Vert {\varvec{\alpha }}_{n} - {\varvec{\alpha }}_{n-1} \Vert _{s}^{\textrm{Lip}(\gamma )}, \ \Vert \breve{{\varvec{\alpha }}}_{n}- \breve{{\varvec{\alpha }}}_{n-1} \Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_n^{2\tau +1} N_{n-1}^{-{\mathfrak {a}}} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \quad n\geqslant 1. \end{aligned} \end{aligned}$$\end{document}

As a consequence, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\varvec{\alpha }}_{n}\Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_0^{2\tau +1} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} . \end{aligned}$$\end{document}

Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}_{n}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\alpha }}}_{n}$$\end{document} are both quasi-periodic traveling waves. In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{0}({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\alpha }}_{n}({\varphi },x)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\alpha }}}_{n}({\varphi },y)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi },y)$$\end{document} ;

(ii) For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} , the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varvec{\alpha }}_{n}({\varphi },x))_{n\in {\mathbb {N}}_0}$$\end{document} (resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\breve{{\varvec{\alpha }}}_{n}({\varphi },y))_{n\in {\mathbb {N}}_0}$$\end{document} ) is a Cauchy sequence with respect to the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \,\cdot \,\Vert _{s}^{\textrm{Lip}(\gamma )}$$\end{document} and it converges to some limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}({\varphi },x)$$\end{document} (resp. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}({\varphi },y)$$\end{document} ), with estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {\varvec{\beta }}- {\varvec{\alpha }}_{n} \Vert _{s}^{\textrm{Lip}(\gamma )} , \ \Vert \breve{{\varvec{\beta }}} - \breve{{\varvec{\alpha }}}_{n}\Vert _{s}^{\textrm{Lip}(\gamma )} \lesssim _{s} N_{n+1}^{2\tau +1} N_{n}^{1-{\mathfrak {a}}} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma }^{\textrm{Lip}(\gamma )} , \\&\quad \Vert {\varvec{\beta }}\Vert _{s}^{\textrm{Lip}(\gamma )}, \ \Vert \breve{{\varvec{\beta }}} \Vert _{s}^{\textrm{Lip}(\gamma )}\lesssim _{s} N_0^{2\tau +1} \varepsilon \gamma ^{-1} \Vert w \Vert _{s+\sigma } . \end{aligned} \end{aligned}$$\end{document}

Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}$$\end{document} are both quasi-periodic traveling waves. In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{0}({\varphi },x)=\textrm{even}({\varphi },x)$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}({\varphi },x)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi },x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}({\varphi },y)$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi },y)$$\end{document} ;

(iii) Define the operators

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {B}}u)({\varphi },x):= h ({\varphi }, x+ {\varvec{\beta }}({\varphi },x)) , \quad ({\mathcal {B}}^{-1} h ) ({\varphi },y):= h ({\varphi }, y + \breve{{\varvec{\beta }}}({\varphi },y)). \end{aligned}$$\end{document}

Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [s_0,S-\sigma ]$$\end{document} , the sequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\widetilde{{\mathcal {B}}}}_{n}^{\pm 1})_{n\in {\mathbb {N}}_0}$$\end{document} converge strongly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s({\mathbb {T}}^{\nu +2})$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}^{\pm 1}$$\end{document} , namely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{n\rightarrow + \infty }\Vert ({\mathcal {B}}^{\pm 1}- {\widetilde{{\mathcal {B}}}}_{n}^{\pm 1})h \Vert _{s} =0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h \in H^s({\mathbb {T}}^{\nu +2})$$\end{document} . Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}^{\pm 1}$$\end{document} are momentum preserving. In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , they are reversibility preserving.

Proof

See the proof of Lemma 4.4 in [7]. The property of the quasi-periodic traveling wave functions and momentum preserving operators follow from Lemma 5.2, since both the properties are closed under limit. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 5.4

The sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathtt m}_{n})_{n\in {\mathbb {N}}_0}$$\end{document} satisfies the bound

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\omega \in {\mathcal {O}}_{n}^\gamma } |{\mathtt m}_{n}(\omega )| \lesssim \varepsilon N_{n-1}^{-{\mathfrak {a}}} . \end{aligned}$$\end{document}

Furthermore, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o} \subseteq \cap _{n \geqslant 0} {\mathcal {O}}_{n}^\gamma $$\end{document} (recalling (3.5), (4.2), (3.6), (5.11)).

Proof

We start with the proof of (5.27). By (5.25), using (5.10) and (5.15), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega \cdot \partial _{{\varphi }} + {\mathtt m}_{n} \cdot \nabla + \textbf{b}_{n} \cdot \nabla = {\widetilde{{\mathcal {T}}}}_{n} = {\widetilde{{\mathcal {B}}}}_{n-1}^{-1} {\widetilde{{\mathcal {T}}}}_{0} {\widetilde{{\mathcal {B}}}}_{n-1} \quad \forall \,\omega \in {\mathcal {O}}_{n}^\gamma , \end{aligned}$$\end{document}

and, by Lemma (5.3)-(i), we compute explicitly

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widetilde{{\mathcal {B}}}}_{n-1}^{-1} {\widetilde{{\mathcal {T}}}}_{0} {\widetilde{{\mathcal {B}}}}_{n-1} = \omega \cdot \partial _{{\varphi }} + {\widetilde{{\mathcal {B}}}}_{n-1}^{-1} \big ( \omega \cdot \partial _{{\varphi }} {\varvec{\alpha }}_{n-1} + \textbf{b}+ \textbf{b}\cdot \nabla {\varvec{\alpha }}_{n-1} \big ) \cdot \nabla . \end{aligned}$$\end{document}

Since, clearly, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{{\mathcal {B}}}}_{n-1} [c]= c$$\end{document} for any constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in {\mathbb {R}}$$\end{document} , we have, by (5.28), (5.29),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt m}_{n} + {\widetilde{{\mathcal {B}}}}_{n-1} \textbf{b}_{n} = \omega \cdot \partial _{{\varphi }} {\varvec{\alpha }}_{n-1}+ \textbf{b}+ \textbf{b}\cdot {\varvec{\alpha }}_{n-1} . \end{aligned}$$\end{document}

We note that, by (4.8), (5.2),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {T}}^{\nu +2}} \omega \cdot \partial _{{\varphi }}{\varvec{\alpha }}_{n-1} \,\textrm{d}{{\varphi }}\,\textrm{d}{x} = 0 , \quad \int _{{\mathbb {T}}^{\nu +2}} \textbf{b}({\varphi },x) \,\textrm{d}{{\varphi }}\,\textrm{d}{x} = 0, \\&\quad \int _{{\mathbb {T}}^{\nu +2}} \textbf{b}\cdot \nabla {\varvec{\alpha }}_{n-1} \,\textrm{d}{{\varphi }}\,\textrm{d}{x} = - \int _{{\mathbb {T}}^{\nu +2}} \textrm{div}(\textbf{b}) {\varvec{\alpha }}_{n-1} \,\textrm{d}{{\varphi }}\,\textrm{d}{x} = 0. \end{aligned} \end{aligned}$$\end{document}

Taking the space-time average of the equation (5.30) and using (5.31), we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt m}_{n} = - \int _{{\mathbb {T}}^{\nu +2}} {\widetilde{{\mathcal {B}}}}_{n-1} \textbf{b}_{n} \,\textrm{d}{{\varphi }}\,\textrm{d}{x} \quad \forall \,\omega \in {\mathcal {O}}_{n}^\gamma . \end{aligned}$$\end{document}

The claimed estimate (5.27) follows by Lemmata 2.2, 2.14 and estimates (5.12), (5.26), (4.2) and (5.4).

We now prove the claim on the inclusion. We argue by induction, that is, we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma ,\tau ) \cap \Lambda _{o} \subseteq {\mathcal {O}}_{n}^\gamma $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}_{n}^\gamma = \Lambda _{o}$$\end{document} , so the claim is trivially satisfied. Now we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma ,\tau ) \cap \Lambda _{o} \subseteq {\mathcal {O}}_{n}^\gamma $$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} and we want to prove the inclusion with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} . By induction hypothesis, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n}^\gamma $$\end{document} . Therefore, by (5.27), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathtt m}_{n}| \lesssim \varepsilon N_{n-1}^{-{\mathfrak {a}}}$$\end{document} . Hence, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,j)\in {\mathbb {Z}}^{\nu +2}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<|\ell | \leqslant N_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell )+j=0$$\end{document} (which implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|j|\lesssim N_{n} $$\end{document} ), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\omega \cdot \ell +{\mathtt m}_{n}(\omega )\cdot j|&\geqslant |\omega \cdot \ell | - |{\mathtt m}_{n}(\omega )\cdot j| \\&\geqslant \frac{2\gamma }{{\langle {\ell }\rangle }^{\tau }} - \varepsilon C N_{n} N_{n-1}^{-{\mathfrak {a}}} \geqslant \frac{\gamma }{{\langle {\ell }\rangle }^\tau } , \end{aligned} \end{aligned}$$\end{document}

for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C N_{n}^{1+\tau } N_{n-1}^{-{\mathfrak {a}}} \varepsilon \gamma ^{-1}\leqslant 1$$\end{document} . This holds for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} , provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C N_0^{1+\tau }\varepsilon \gamma ^{-1} \leqslant 1$$\end{document} , which is satisfied by the smallness condition (5.4) and by (5.9). Thus, by the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}_{n+1}^\gamma $$\end{document} (see (5.11) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} ), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal {O}}_{n+1}^\gamma $$\end{document} . The proof is concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 5.1

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , by Lemma 5.4, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt m}_{n}\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document} . By (5.26), (5.12) and Lemma 2.2, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\widetilde{{\mathcal {B}}}}_{n-1} \textbf{b}_{n} \Vert _{s_0} \lesssim \Vert \textbf{b}_{n}\Vert \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document} . Moreover,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \partial _{{\varphi }} {\varvec{\alpha }}_{n-1} - \partial _{{\varphi }}{\varvec{\beta }}\Vert _{s_0}, \Vert \nabla {\varvec{\alpha }}_{n-1}-\nabla {\varvec{\beta }}\Vert _{s_0} \leqslant \Vert {\varvec{\alpha }}_{n-1}-{\varvec{\beta }}\Vert _{s_0+1} \rightarrow 0 \quad \text {as} \ \ n\rightarrow \infty . \end{aligned}$$\end{document}

Hence, passing to the limit in the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \,\cdot \,\Vert _{s_0}$$\end{document} in identity (5.30), we obtain the identity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega \cdot \partial _{{\varphi }} {\varvec{\beta }}+ \textbf{b}({\varphi },x) + \textbf{b}({\varphi },x) \cdot {\varvec{\beta }}=0 \end{aligned}$$\end{document}

in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s_0}({\mathbb {T}}^{\nu +1})$$\end{document} and therefore pointwise for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\varphi },x) \in {\mathbb {T}}^{\nu +2}$$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} . As a consequence,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {B}}^{-1} {\widetilde{{\mathcal {T}}}}{\mathcal {B}}= \omega \cdot \partial _{{\varphi }} + \big \{ {\mathcal {B}}^{-1} \big ( \omega \cdot \partial _{{\varphi }}{\varvec{\beta }}+ \textbf{b}+ \textbf{b}\cdot \nabla {\varvec{\beta }}\big ) \big \} \cdot \nabla = \omega \cdot \partial _{{\varphi }} \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , which shows (5.6). Estimates (5.5), (5.7) follow by Lemma 5.3-(ii) and Lemma 2.2.

It remains to prove the estimates (5.8). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{i}:= \textbf{b}_{i}(w_i)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} satisfy (4.8) and assume that, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 > s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1,w_2$$\end{document} satisfy (4.2), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0 \geqslant s_1 + \sigma $$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}_{i}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{i}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} , be the corresponding functions and operators given by Lemma (5.3)-(ii),(iii). Then, by (5.32) and (5.6), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} , we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {B}}_{i}^{-1} (\omega \cdot \partial _{{\varphi }} + \textbf{b}_{i} \cdot \nabla ) {\mathcal {B}}_{i} = \omega \cdot \partial _{{\varphi }}, \quad (\omega \cdot \partial _{{\varphi }} + \textbf{b}_{i}\cdot \nabla )[{\varvec{\beta }}_{i}] + \textbf{b}_{i} = 0. \end{aligned}$$\end{document}

Hence, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}{\varvec{\beta }}:= {\varvec{\beta }}_{1}(w_1)-{\varvec{\beta }}_{2}(w_2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}\textbf{b}:= \textbf{b}_{1}(w_1)-\textbf{b}_{2}(w_2)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\omega \cdot \partial _{{\varphi }} + \textbf{b}_{1}(w_1) \cdot \nabla )[\Delta _{12}{\varvec{\beta }}] + \mathfrak {g}= 0, \quad \mathfrak {g}:= \Delta _{12}\textbf{b}\cdot \nabla \textbf{b}_{2}(w_2) + \Delta _{12}\textbf{b}. \end{aligned}$$\end{document}

By (5.33), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \cdot \partial _{{\varphi }} + \textbf{b}_{1}(w_1)\cdot \nabla = {\mathcal {B}}_{1}^{-1} \omega \cdot \partial _{{\varphi }} {\mathcal {B}}_{1}$$\end{document} , and therefore

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega \cdot \partial _{{\varphi }} {\mathcal {B}}_{1}^{-1}\Delta _{12}{\varvec{\beta }}+ {\mathcal {B}}_{1}^{-1} \mathfrak {g}= 0. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle {{\mathcal {B}}_{1}^{-1} \mathfrak {g}}\rangle }_{{\varphi },x} = - {\langle {\omega \cdot \partial _{{\varphi }} {\mathcal {B}}_{1}^{-1}\Delta _{12}{\varvec{\beta }}}\rangle }_{{\varphi },x}=0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}{\varvec{\beta }}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{1}^{-1}\Delta _{12}{\varvec{\beta }}= \textrm{odd}({\varphi },x)$$\end{document} (by Lemma 5.3-(ii),(iii)), one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _{12}{\varvec{\beta }}= - {\mathcal {B}}_{1} (\omega \cdot \partial _{{\varphi }})^{-1} {\mathcal {B}}_{1}^{-1} \mathfrak {g}, \quad {\forall \,\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)} . \end{aligned}$$\end{document}

Then, having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert w_{i}\Vert _{s_1+\sigma }\leqslant 1$$\end{document} , by (5.7), (5.5) and Lemma 2.2, recalling also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}_{i} = \varepsilon {\mathfrak {B}}[w_i]$$\end{document} by (5.2), (4.4), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} , we get the estimate in (5.8) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}{\varvec{\beta }}$$\end{document} . The corresponding estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}\breve{{\varvec{\beta }}}$$\end{document} can be proved by using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\breve{{\varvec{\beta }}}_{i}=-{\mathcal {B}}_{i}^{-1}{\varvec{\beta }}_{i}$$\end{document} and the mean value theorem. Finally, the estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}{\mathcal {B}}^{\pm 1}$$\end{document} follow by the estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}{\varvec{\beta }}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12}\breve{{\varvec{\beta }}}$$\end{document} , the mean value theorem and Lemma 2.2. The proof is concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now compute the complete conjugation of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}: H_0^{s+1}({\mathbb {T}}^{\nu +2})\rightarrow H_0^s({\mathbb {T}}^{\nu +2})$$\end{document} in (4.3) by means of the transformation in (5.1), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\beta }}({\varphi },x)$$\end{document} as in Proposition 5.1.

Proposition 5.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S>s_0+\sigma _{1}$$\end{document} , for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{1}=\sigma _{1}(\tau ,\nu ) \gg \sigma $$\end{document} (where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} is provided by Proposition 5.1). There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta (S,\tau ,\nu )\in (0,1)$$\end{document} small enough such that, if (4.2) and (5.4) are fulfilled, the following holds:

(i) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{\perp }:= \Pi _0^{\perp } {\mathcal {B}}\Pi _0^{\perp }$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _0^{\perp }$$\end{document} is defined in (2.17) and the invertible map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}$$\end{document} , with inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}^{-1}$$\end{document} , is constructed in Proposition 5.1. Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{0} \leqslant s\leqslant S-\sigma $$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} as in Proposition (5.1), the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{\perp }: H^s_0({\mathbb {T}}^{\nu + 2}) \rightarrow H^s_0({\mathbb {T}}^{\nu + 2})$$\end{document} is invertible with bounded inverse given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{\perp }^{-1}:= \Pi _0^\perp {\mathcal {B}}^{-1}\Pi _0^\perp : H_0^{s}({\mathbb {T}}^{\nu +2})\rightarrow H_{0}^{s}({\mathbb {T}}^{\nu +2})$$\end{document} ;

(ii) For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , as in (3.5), one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{1}: ={\mathcal {B}}_{\perp }^{-1} {\mathcal {L}}\, {\mathcal {B}}_{\perp } = \lambda \, \omega \cdot \partial _{{\varphi }} + \beta \,{\mathtt L}+ {\mathcal {E}}_{1}, \end{aligned}$$\end{document}

where, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _1$$\end{document} ,

the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} satisfies the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | {\mathcal {E}}_{1} |_{-1,s}^{\textrm{Lip}(\gamma )} \lesssim _{s} \lambda ^{\theta } \Vert w \Vert _{s + \sigma _1}^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 \geqslant s_0$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1, w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma _{0} \geqslant s_1 + \sigma _{1} $$\end{document} . Then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&| \Delta _{12} {\mathcal {E}}_{1} |_{-1,s_1} \lesssim _{s_1} \lambda ^{\theta } \Vert w_1 -w_2 \Vert _{s_1 +\sigma _{1}}. \end{aligned} \end{aligned}$$\end{document}

Furthermore, the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} are momentum preserving and, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)= \textrm{odd}({\varphi },x)$$\end{document} , they are reversibility preserving.

Proof

Item (i) is proved in Lemma 4.2 in [27]. We now prove item (ii). By (4.3), (5.2), we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}= {\mathcal {T}}+ \beta \,{\mathtt L}+ \lambda ^{\theta } {\mathcal {E}}_{0}$$\end{document} . We have to compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {B}}_{\perp }^{-1} {\mathcal {L}}\,{\mathcal {B}}_{\perp }&= {\mathcal {B}}_{\perp }^{-1} {\mathcal {T}}{\mathcal {B}}_{\perp } + \beta \, {\mathcal {B}}_{\perp }^{-1} {\mathtt L}\, {\mathcal {B}}_{\perp } +\lambda ^{\theta } {\mathcal {B}}_{\perp }^{-1} {\mathcal {E}}_{0}\, {\mathcal {B}}_{\perp } . \end{aligned} \end{aligned}$$\end{document}

By Proposition 5.1, by item (i) and by (4.10), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {B}}_{\perp }^{-1} {\mathcal {T}}{\mathcal {B}}_{\perp } = \Pi _0^\perp {\mathcal {B}}^{-1} {\mathcal {T}}{\mathcal {B}}\Pi _0^{\perp } = \Pi _0^{\perp } \,\lambda \,\omega \cdot \partial _{{\varphi }} \Pi _0^{\perp } = \lambda \,\omega \cdot \partial _{{\varphi }} \textrm{Id}_0 \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Id}_0$$\end{document} is the identity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_0^2({\mathbb {T}}^{\nu +2})$$\end{document} . To compute the other terms, the main tool is Lemma 2.16-(ii), which provides the structure for the conjugation of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{-1}$$\end{document} as in (2.19). Recalling by (1.5) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt L}=-(-\Delta )^{-1} \partial _{x_1}$$\end{document} , which is invariant on the functions with zero average, we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {B}}_{\perp }^{-1} {\mathtt L}\, {\mathcal {B}}_{\perp }&= - \Pi _0^{\perp }\big ( {\mathcal {B}}^{-1} (-\Delta )^{-1} {\mathcal {B}}\big ) \Pi _0^{\perp } \big ( {\mathcal {B}}^{-1} \partial _{x_1} {\mathcal {B}}\big )\Pi _0^\perp \\&= - \big ( (-\Delta )^{-1} + {\mathcal {P}}_{-2} \big ) \Pi _0^\bot \{ {\mathcal {B}}^{-1}( 1+ {\varvec{\beta }}_{x_1} )\} \partial _{y_1} \\&= {\mathtt L}- (-\Delta )^{-1} \Pi _0^\perp \{{\mathcal {B}}^{-1}({\varvec{\beta }}_{x_1})\} \partial _{y_1} - {\mathcal {P}}_{-2}\, \Pi _0^\perp \{{\mathcal {B}}^{-1}( 1+ {\varvec{\beta }}_{x_1} )\} \partial _{y_1} . \end{aligned} \end{aligned}$$\end{document}

In order to conjugate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{0}$$\end{document} we note that, by its definition in (4.4) and by (1.8), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {E}}_{0} = \nabla w \cdot {\mathfrak {B}}= w_{x_1} (-\Delta )^{-1} \partial _{x_2} - w_{x_2} (-\Delta )^{-1} \partial _{x_1}. \end{aligned}$$\end{document}

Using the same computation in (5.37), we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}_{\perp }^{-1} (-\Delta )^{-1} \partial _{x_m} {\mathcal {B}}_{\perp } \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=1,2$$\end{document} , and consequently, using also (4.9), (4.10),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {B}}_{\perp }^{-1} {\mathcal {E}}_{0}\,{\mathcal {B}}_{\perp }&= \Pi _0^\perp \{{\mathcal {B}}^{-1} ( w_{x_1})\} {\mathcal {B}}_{\perp }^{-1} (-\Delta )^{-1} \partial _{x_2} {\mathcal {B}}_{\perp } - \Pi _0^\perp \{{\mathcal {B}}^{-1} ( w_{x_2})\} {\mathcal {B}}_{\perp }^{-1} (-\Delta )^{-1} \partial _{x_1} {\mathcal {B}}_{\perp }\nonumber \\&= \Pi _0^\perp \{{\mathcal {B}}^{-1}(\nabla w)\} \cdot {\mathcal {B}}_{\perp }^{-1} (-\Delta )^{-1} \nabla ^\perp {\mathcal {B}}_{\perp } \end{aligned}$$\end{document}

is in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} . We conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}:= {\mathcal {B}}_{\perp }^{-1} {\mathcal {L}}\,{\mathcal {B}}_{\perp }$$\end{document} has the form (5.34), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} , recalling (5.37), (5.38), is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{1}&:= \beta \big ( { -(-\Delta )^{-1} \Pi _0^\perp \{{\mathcal {B}}^{-1}({\varvec{\beta }}_{x_1})\} \partial _{y_1} - {\mathcal {P}}_{-2}\, \Pi _0^\perp \{{\mathcal {B}}^{-1}( 1+ {\varvec{\beta }}_{x_1} )\} \partial _{y_1} } \big ) \\&\quad + \lambda ^{\theta } \Pi _0^\perp \{{\mathcal {B}}^{-1}(\nabla w)\} \cdot {\mathcal {B}}_{\perp }^{-1} (-\Delta )^{-1} \nabla ^\perp {\mathcal {B}}_{\perp } . \end{aligned} \end{aligned}$$\end{document}

The estimate (5.35) follows by (5.39), Lemma 2.2, Lemma 2.6-(ii), estimate (2.18), Lemma 2.16 and Proposition 5.1 with the smallness condition (5.4). The estimate (5.36) follows by similar arguments from the explicit expression (5.39) and we omit the details. Finally, by Lemma 2.9, 2.11 and Proposition 5.1, we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} is real and momentum preserving. If w is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_1$$\end{document} is also reversible. This concludes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Reduction to perturbative of the large remainder

The next goal is to conjugate the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} in (5.34) through a series of transformation in order to reduce both the size and the order of the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-1}$$\end{document} . We shall prove the following Proposition.

Proposition 5.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\in \ {\mathbb {N}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \frac{1-{\mathtt c}}{2(1-{\mathtt c}) - \alpha }$$\end{document} . There exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _M= \sigma _{M}(\tau ,\nu ,M) \gg 0$$\end{document} large enough such that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S>s_0 +\sigma _{M}$$\end{document} , there is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \equiv \delta (S, M) \ll 1$$\end{document} small enough such that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \leqslant \delta $$\end{document} and if (4.2) is fullfilled with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{0} = \sigma _M$$\end{document} , then the following hold. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , there exists a real, invertible operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Phi }_M \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _M$$\end{document} satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\mathbf{\Phi }_M^{\pm 1}|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, M} 1 + \Vert {w} \Vert _{s + \sigma _M}^{\textrm{Lip}(\gamma )} , \quad \forall s_0 \leqslant s \leqslant S - \sigma _M, \\&\quad |\mathbf{\Phi }_M^{\pm 1}-\textrm{Id}|_{-M, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, M} \Vert {w} \Vert _{s + \sigma _M}, \, \end{aligned} \end{aligned}$$\end{document}

such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {L}}}_M:= \mathbf{\Phi }_M^{- 1} {{\mathcal {L}}}_1 \mathbf{\Phi }_M = \lambda \, \omega \cdot \partial _\varphi + \beta \,{\mathtt L}+ {{\mathcal {Z}}}_M + {{\mathcal {E}}}_M, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_M \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- 1}_s$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant 0$$\end{document} , is a diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_M:= \textrm{diag}_{j \ne 0} z_M(j)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_M \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- M}_s$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _M$$\end{document} satisfy the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {Z}}}_M|_{- 1, s}^{\textrm{Lip}(\gamma )} \lesssim _{M} \lambda ^{\theta }, \quad \forall s \geqslant s_0, \quad \sup _{j \ne 0} |j| |z_M(j)|^{\textrm{Lip}(\gamma )} \lesssim _M \lambda ^{\theta }, \\&\quad |{{\mathcal {E}}}_M|_{- M, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, M} \big ( \lambda ^{\theta -1}\gamma ^{-1} \big )^{M-1} \lambda ^{\theta } \Vert {w} \Vert _{s + \sigma _M}^{\textrm{Lip}(\gamma )} , \quad \forall s_0 \leqslant s \leqslant S - \sigma _M. \end{aligned} \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 \geqslant s_0$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w_1, w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma _{0} \geqslant s_1 + \sigma _{M} $$\end{document} . Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} , one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Delta _{12} \mathbf{{\Phi }}_{M}^{\pm 1} |_{-M,s_1} \lesssim _{s_1, M} \Vert w_1 - w_2 \Vert _{s_1 + \sigma _{M}},\\&\quad | \Delta _{12} {\mathcal {E}}_{M}|_{-M,s_1}\lesssim _{s_1, M} \big ( \lambda ^{\theta -1}\gamma ^{-1} \big )^{M-1} \lambda ^{\theta } \Vert w_1 - w_2 \Vert _{s_1 + \sigma _{M}}. \end{aligned} \end{aligned}$$\end{document}

Furthermore, the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{M}^{\pm 1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{M}$$\end{document} are real and momentum preserving. In addition, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{M}^{\pm 1}$$\end{document} are reversibility preserving and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{M}$$\end{document} is reversible.

Proposition 5.6 is proved as a consequence of the following iterative procedure.

Lemma 5.7

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\in \ {\mathbb {N}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \frac{1-{\mathtt c}}{2(1-{\mathtt c}) - \alpha } $$\end{document} . There exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _1< \sigma _2< \ldots < \sigma _M$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{i}:=\sigma _{i}(\tau ,\nu )>0$$\end{document} large enough such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S>s_0 +\sigma _{M}$$\end{document} , there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \equiv \delta (S, M) \ll 1$$\end{document} small enough such that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \leqslant \delta $$\end{document} and if (4.2) is fullfilled with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{0} = \sigma _M$$\end{document} , then the following holds. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 1, \ldots , M$$\end{document} , there exists a real operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_m$$\end{document} of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {L}}}_m = \lambda \, \omega \cdot \partial _\varphi + \beta \,{\mathtt L}+ {{\mathcal {Z}}}_m + {{\mathcal {E}}}_m, \end{aligned}$$\end{document}

where, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _m$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_m \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- 1}_s$$\end{document} is the diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_m = \textrm{diag}_{j \ne 0} z_m(j)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_m \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- m}_s$$\end{document} and they satisfy the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {Z}}}_m|_{- 1, s}^{\textrm{Lip}(\gamma )} \lesssim _m \lambda ^{\theta } , \quad \sup _{j \ne 0} |j| |z_m(j)|^{\textrm{Lip}(\gamma )} \lesssim _m \lambda ^{\theta },\\&\quad |{{\mathcal {E}}}_m|_{- m, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{m - 1} \lambda ^{\theta } \Vert {w} \Vert _{s + \sigma _m}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

There exist real operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \Phi _{m} \}_{m=1}^{M - 1}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{m}:= \textrm{exp} ({\mathcal {X}}_{m})\in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{0}$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {X}}_{m}:= {\mathcal {X}}_{m}({\varphi }) = \sum _{\ell \in {\mathbb {Z}}^{\nu }\setminus \{0\}} {\widehat{{\mathcal {X}}}}_{m}(\ell )e^{\textrm{i}\,\ell \cdot {\varphi }} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-m}, \end{aligned}$$\end{document}

satisfying the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{\mathcal {X}}_m|_{- m, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} \big (\lambda ^{\theta -1} \gamma ^{- 1} \big )^m \Vert {w} \Vert _{s + \sigma _m}^{\textrm{Lip}(\gamma )}, \quad \forall s_0 \leqslant s \leqslant S - \sigma _m, \\&\quad |\Phi _m^{\pm 1}|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} 1 + \Vert {w} \Vert _{s + \sigma _{m}}^{\textrm{Lip}(\gamma )}, \quad \forall s_0 \leqslant s \leqslant S - \sigma _m. \end{aligned} \end{aligned}$$\end{document}

Moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 2, \ldots , M - 1$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{m}:= \Phi _{m-1}^{-1} {\mathcal {L}}_{m-1} \Phi _{m-1} . \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 \geqslant s_0$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1,w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma _{0} \geqslant s_1 + \sigma _{m} $$\end{document} . Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}(w_1)\cap \Lambda _{o}(w_2)$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Delta _{12} \Phi _{m}^{\pm 1} |_{-m,s_1} \lesssim _{s_1, m} \Vert w_1 - w_2 \Vert _{s_1 + \sigma _{m}} ,\\&\quad | \Delta _{12} {\mathcal {E}}_{m}|_{-m,s_1}\lesssim _{s_1, m} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{m - 1} \lambda ^{\theta } \Vert w_1 - w_2 \Vert _{s_1 + \sigma _{m}}. \end{aligned} \end{aligned}$$\end{document}

Furthermore, the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{m}^{\pm 1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{m}$$\end{document} are real and momentum preserving. In addition, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{m}^{\pm 1}$$\end{document} are reversibility preserving and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{m}$$\end{document} is reversible.

Proof

We proceed by induction. The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} in (5.34) is of the form (5.44) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}_{1}=0$$\end{document} . By (5.35) and (5.36) we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document} satisfies (5.45) and (5.48). Finally, by Proposition 5.5 we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{1}$$\end{document} is reversible and momentum preserving.

We now assume that the claimed statements hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 1, \ldots , M - 1$$\end{document} and we prove them for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m + 1$$\end{document} . We look for a trasformation of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi _{m}:= \textrm{exp}({\mathcal {X}}_m) \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{0}, \quad {\mathcal {X}}_{m}:= {\mathcal {X}}_{m}({\varphi }) = \sum _{\ell \in {\mathbb {Z}}^\nu \setminus \{0\}} {\widehat{{\mathcal {X}}}}_{m}(\ell ) e^{\textrm{i}\,\ell \cdot {\varphi }} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{-m}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\mathcal {X}}}}_{m}$$\end{document} has to be determined. To compute \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_{m + 1} = \Phi _m^{- 1} {{\mathcal {L}}}_m \Phi _m$$\end{document} , we look separately at the conjugation of the three terms appearing in (5.44). By the standard Lie expansion, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Phi _m^{- 1} \lambda \, \omega \cdot \partial _\varphi \Phi _m&= \lambda \, \omega \cdot \partial _\varphi + \lambda \, \omega \cdot \partial _\varphi {\mathcal {X}}_m(\varphi ) + {{\mathcal {Q}}}_m^{(1)}(\varphi ) , \\ {{\mathcal {Q}}}_m^{(1)}(\varphi )&= \int _0^1 (1 - \tau ) \textrm{exp}(- \tau {\mathcal {X}}_m) [\lambda \, \omega \cdot \partial _\varphi {\mathcal {X}}_m(\varphi ), {\mathcal {X}}_m(\varphi )] \textrm{exp}( \tau {\mathcal {X}}_m) \,\textrm{d}{\tau }, \\ \Phi _m^{- 1} ( \beta {\mathtt L}) \Phi _m&= \beta {\mathtt L}+ {{\mathcal {Q}}}_m^{(2)}, \\ {{\mathcal {Q}}}_m^{(2)}(\varphi )&:= \int _0^1 \textrm{exp}(- \tau {\mathcal {X}}_m(\varphi )) [\beta {\mathtt L}, {\mathcal {X}}_m(\varphi )] \textrm{exp}(\tau {\mathcal {X}}_m(\varphi )) \,\textrm{d}{\tau }, \\ \Phi _m^{- 1} {{\mathcal {Z}}}_m \Phi _m&= {{\mathcal {Z}}}_m + {{\mathcal {Q}}}_m^{(3)}, \\ {{\mathcal {Q}}}_m^{(3)}(\varphi )&:= \int _0^1 \textrm{exp}(- \tau {\mathcal {X}}_m(\varphi )) [{{\mathcal {Z}}}_m, {\mathcal {X}}_m(\varphi )] \textrm{exp}(\tau {\mathcal {X}}_m(\varphi )) \,\textrm{d}{\tau }, \\ \Phi _m^{- 1} {{\mathcal {E}}}_m \Phi _m&= {{\mathcal {E}}}_m + {{\mathcal {Q}}}_m^{(4)}, \\ {{\mathcal {Q}}}_m^{(4)}(\varphi )&:= \int _0^1 \textrm{exp}(- \tau {\mathcal {X}}_m(\varphi )) [{{\mathcal {E}}}_m(\varphi ), {\mathcal {X}}_m(\varphi )] \textrm{exp}(\tau {\mathcal {X}}_m(\varphi )) \,\textrm{d}{\tau }.\\ \end{aligned} \end{aligned}$$\end{document}

The terms of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- m$$\end{document} in the expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_{m + 1}= \Phi _m^{- 1} {{\mathcal {L}}}_m \Phi _m$$\end{document} that we want to reduce are then given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \, \omega \cdot \partial _\varphi {\mathcal {X}}_m(\varphi ) + {{\mathcal {E}}}_m(\varphi )$$\end{document} . Hence, we solve the homological equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \, \omega \cdot \partial _\varphi {\mathcal {X}}_m(\varphi ) + {\mathcal E}_m(\varphi ) = \widehat{{\mathcal {E}}}_m(0), \quad \widehat{\mathcal E}_m(0):= \frac{1}{(2 \pi )^\nu } \int _{{\mathbb {T}}^\nu } {{\mathcal {E}}}_m(\varphi ) \,\textrm{d}{\varphi }, \end{aligned}$$\end{document}

by defining, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {X}}}_m(\varphi ) = - \sum _{\ell \ne 0} \frac{\widehat{\mathcal E}_m(\ell )}{\textrm{i}\lambda \, \omega \cdot \ell } e^{\textrm{i}\ell \cdot \varphi }. \end{aligned}$$\end{document}

Hence the matrix elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {X}}}_m$$\end{document} are given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \widehat{{\mathcal {X}}}_m(\ell )_j^{j'} = {\left\{ \begin{array}{ll} - \dfrac{\widehat{{\mathcal {E}}}_m(\ell )_j^{j'}}{\textrm{i}\lambda \,\omega \cdot \ell }, & \ell \ne 0, \quad j, j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}, \quad \pi ^\top (\ell ) + j - j' = 0, \\ 0 & \text {otherwise.} \end{array}\right. } \end{aligned}$$\end{document}

Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , one has that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\widehat{{\mathcal {X}}}_m(\ell )_j^{j'}| \lesssim \gamma ^{- 1} \lambda ^{- 1} \langle \ell \rangle ^\tau |\widehat{\mathcal E}_m(\ell )_j^{j'}| \end{aligned}$$\end{document}

and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1, \omega _2 \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\widehat{{\mathcal {X}}}_m(\ell ; \omega _1)_j^{j'} - \widehat{{\mathcal {X}}}_m(\ell ; \omega _2)_j^{j'}|&\lesssim \langle \ell \rangle ^{2 \tau + 1} \gamma ^{- 2} \lambda ^{- 1} |\widehat{{\mathcal {E}}}_m(\ell ; \omega _2)_j^{j'}| |\omega _1 - \omega _2| \\&\quad + \langle \ell \rangle ^\tau \gamma ^{- 1} \lambda ^{- 1} |\widehat{{\mathcal {E}}}_m(\ell ; \omega _1)_j^{j'} - \widehat{\mathcal E}_m(\ell ; \omega _2)_j^{j'}|. \end{aligned} \end{aligned}$$\end{document}

By taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma _{m} + 2 \tau + 1$$\end{document} , the latter two estimates, together with the Definition 2.5 imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\mathcal {X}}}_m \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^{- m}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _{m} - 2 \tau - 1$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {X}}}_m|_{- m, s}^{\textrm{Lip}(\gamma )} \lesssim _s \gamma ^{- 1} \lambda ^{- 1} |{{\mathcal {E}}}_m|_{- m, s + 2 \tau + 1}^{\textrm{Lip}(\gamma )} \\&\qquad \qquad {\mathop {\lesssim _{s, m}}\limits ^{(5.7)}} \lambda ^{- 1} \gamma ^{- 1} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{m - 1} \lambda ^{\theta } \Vert {w} \Vert _{s + \sigma _m + 2 \tau + 1}^{\textrm{Lip}(\gamma )} \\&\qquad \qquad \lesssim _{s, m} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{m } \Vert {w} \Vert _{s + \sigma _m + 2 \tau + 1}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

Together with Lemma 2.6-(iv) and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \ll 1$$\end{document} , we have that (5.51) implies, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _m - 2 \tau - 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\tau \in [- 1, 1]}|\textrm{exp}(\tau {\mathcal {X}}_m)|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} 1 + \Vert {w} \Vert _{s + \sigma _m + 2 \tau + 1}^{\textrm{Lip}(\gamma )} . \end{aligned}$$\end{document}

By (5.49), we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {{\mathcal {L}}}_{m + 1}&= \Phi _m^{- 1} {{\mathcal {L}}}_m \Phi _m = \lambda \,\omega \cdot \partial _\varphi + \beta \, {\mathtt L}+ {{\mathcal {Z}}}_{m + 1} + {{\mathcal {E}}}_{m + 1} , \\ {{\mathcal {Z}}}_{m + 1}&:= {{\mathcal {Z}}}_m + \widehat{{\mathcal {E}}}_m(0), \\ {{\mathcal {E}}}_{m + 1}&:= {{\mathcal {Q}}}_m^{(1)} + {\mathcal Q}_m^{(2)} + {{\mathcal {Q}}}_m^{(3)} + {{\mathcal {Q}}}_m^{(4)}. \end{aligned} \end{aligned}$$\end{document}

Properties and estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_{m + 1}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_m$$\end{document} is momentum preserving, by Lemma 2.13 we have that the time-independent operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{{\mathcal {E}}}(0)$$\end{document} is diagonal and hence also the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_{m+1} = \textrm{diag}_{j \ne 0} z_{m+1}(j)$$\end{document} is a diagonal operator with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{m + 1}(j):= z_m(j) + \widehat{\mathcal E}_m(0)_j^{j}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} . Furthermore, by Lemma 2.6-(v) and by the induction estimates (5.45) (using also the ansatz (4.2)), we get that the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_{m + 1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- 1}_s$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant s_0$$\end{document} and it satisfies, using also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \ll 1$$\end{document} by (4.7),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |{{\mathcal {Z}}}_{m + 1}|_{- 1, s}^{\textrm{Lip}(\gamma )}&\lesssim |{{\mathcal {Z}}}_m|_{- 1, s}^{\textrm{Lip}(\gamma )} + |\widehat{{\mathcal {E}}}_m(0)|_{- m, s} \lesssim |{{\mathcal {Z}}}_m|_{- 1, s}^{\textrm{Lip}(\gamma )} + |{{\mathcal {E}}}_m|_{- m, s_0}^{\textrm{Lip}(\gamma )} \\&\lesssim _m \lambda ^{\theta } + \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m - 1} \lambda ^{\theta } \lesssim _m \lambda ^{\theta }, \end{aligned} \end{aligned}$$\end{document}

which implies that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{j \ne 0} |j| |z_{m + 1}(j)|^{\textrm{Lip}(\gamma )} \lesssim _m \lambda ^{\theta }. \end{aligned}$$\end{document}

Properties and estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_{m + 1}$$\end{document} . We now estimate the remainder \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_{m + 1}$$\end{document} in (5.53). We have to analyze the four terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Q}}}_m^{(1)}, {{\mathcal {Q}}}_m^{(2)}, {{\mathcal {Q}}}_m^{(3)}, {{\mathcal {Q}}}_m^{(4)}$$\end{document} in (5.49). First, by (5.50), we note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {Q}}}_m^{(1)} = \int _0^1 (1 - \tau ) \textrm{exp}(- \tau {\mathcal {X}}_m) { [{{\mathcal {Z}}}_{m + 1}-{{\mathcal {Z}}}_{m} - {{\mathcal {E}}}_m, {\mathcal {X}}_m] } \textrm{exp}( \tau {\mathcal {X}}_m) \,\textrm{d}{\tau }. \end{aligned}$$\end{document}

By the estimates (5.45), (5.51), (5.54), using also (2.21) Lemma 2.6-(ii) the ansatz (4.2), we get that, taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma _m + m + 2 \tau + 1$$\end{document} , the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [{{\mathcal {Z}}}_{m + 1}, {\mathcal {X}}_m] \,, {[{{\mathcal {Z}}}_{m}, {\mathcal {X}}_m] } \,,[{\mathtt L}, {\mathcal {X}}_m] \,, \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^{- m - 1}, [{\mathcal E}_m, {\mathcal {X}}_m] \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^{- 2\,m} {\subseteq } {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}^{- m - 1}_s$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ s_0 \leqslant s \leqslant S - \sigma _m - m- 2 \tau - 1$$\end{document} and they satisfy the following estimates, recalling also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \ll 1$$\end{document} by (4.7):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} | [{{\mathcal {Z}}}_{m + 1}, {\mathcal {X}}_m]|_{- m - 1, s}^{\textrm{Lip}(\gamma )}&\lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^m \Vert {w} \Vert _{s + \sigma _m + m + 2 \tau + 1}^{\textrm{Lip}(\gamma )}, \\ { | [{{\mathcal {Z}}}_{m}, {\mathcal {X}}_m]|_{- m - 1, s}^{\textrm{Lip}(\gamma )} }&\lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^m \Vert {w} \Vert _{s + \sigma _m + m + 2 \tau + 1}^{\textrm{Lip}(\gamma )}, \\ |[{{\mathcal {E}}}_m, {\mathcal {X}}_m]|_{- m - 1, s}^{\textrm{Lip}(\gamma )}&\leqslant |[{{\mathcal {E}}}_m, {\mathcal {X}}_m]|_{- 2m, s}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m - 1} \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m} \Vert {w} \Vert _{s + \sigma _m + m + 2 \tau + 1}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m} \Vert {w} \Vert _{s + \sigma _m + m + 2 \tau + 1}^{\textrm{Lip}(\gamma )}, \\ |[{\mathtt L}, {\mathcal {X}}_m]|_{- m - 1, s}^{\textrm{Lip}(\gamma )}&\lesssim _{s, m} \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m} \Vert {w} \Vert _{s + \sigma _m + 2 \tau + 1}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1}\big )^{m} \Vert {w} \Vert _{s + \sigma _m + 2 \tau + 1}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

Together with (5.52), and Lemma 2.6-(ii), the latter estimates imply that, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{m + 1} \geqslant \sigma _m + m + 2 \tau + 2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {w} \Vert _{s_0 + \sigma _{m + 1}}^{\textrm{Lip}(\gamma )} \lesssim 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \sigma _{m + 1}$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_{m + 1} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_s^{- m - 1}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _{m + 1}$$\end{document} and it satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{{\mathcal {E}}}_{m + 1}|_{- m - 1, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, m} \lambda ^{\theta } \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^m \Vert {w} \Vert _{s + \sigma _{m + 1}}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

The estimates in (5.48) follow by similar arguments and we omit the details. By Lemma 2.11 and the inductive hypotheses on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{m}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{m}$$\end{document} , we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{m+1}, {{\mathcal {L}}}_{m + 1}$$\end{document} are momentum preserving. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} then by Lemma 2.9, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _m$$\end{document} is reversibility preserving and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_{m + 1}, {{\mathcal {E}}}_{m + 1}, {{\mathcal {Z}}}_{m + 1}$$\end{document} are reversible operators. The proof of the claimed statements at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m + 1$$\end{document} is then concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 5.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Phi }_{1}:=\textrm{Id}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{\Phi }_M:= \Phi _{1} \circ \Phi _{2} \circ \ldots \circ \Phi _{M - 1}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\geqslant 2$$\end{document} . By Lemma 5.7, Lemma 2.6-(ii) and the estimates (5.46), (5.48), we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{M}$$\end{document} satisfies the estimates (5.40), (5.43). Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , then the conjugation (5.41) holds with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Z}}}_M$$\end{document} satisfying the estimate in (5.42), using also Lemma 2.6-(v), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_M$$\end{document} satisfying the estimate, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \sigma _M$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{{\mathcal {E}}}_M|_{- M, s}^{\textrm{Lip}(\gamma )} \lesssim _{s, M} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{M - 1} \lambda ^{\theta } \Vert {w} \Vert _{s + \sigma _M}^{\textrm{Lip}(\gamma )} . \end{aligned}$$\end{document}

Then, the desired bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}_M$$\end{document} in (5.42) follows since, using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \frac{1-{\mathtt c}}{2(1-{\mathtt c}) - \alpha } = \frac{1-{\mathtt c}}{1-{\mathtt c}-\theta } $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough, and recalling (4.5), (4.6), we have, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\lambda ^{-{\mathtt c}}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big ( \lambda ^{\theta -1} \gamma ^{- 1} \big )^{M - 1} \lambda ^{\theta } = \lambda ^{M\theta -(M-1)(1-{\mathtt c})} < 1. \end{aligned}$$\end{document}

Finally, by Lemma 2.11 and Proposition 5.5, we obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{M}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal L}_M$$\end{document} are momentum preserving. Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} , then by Lemma 2.9, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{{\Phi }}_{M}$$\end{document} is reversibility preserving and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_M$$\end{document} is reversible. The proof of the Proposition is then concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

KAM perturbative reduction

We are now in position to reduce the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{M}$$\end{document} in (5.47) with a KAM reducibility iteration. To the purposes of the KAM reducibility, we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{L}_{0}: ={\mathcal {L}}_{M} = \lambda \,\omega \cdot \partial _{{\varphi }} + \textbf{D}_{0} + \textbf{E}_{0} \end{aligned}$$\end{document}

where:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} The diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_{0}$$\end{document} is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\textbf{D}_{0}:= \beta \,{\mathtt L}+ \textbf{Z}_{0} = {{\,\textrm{diag}\,}}_{j\in {\mathbb {Z}}^2\setminus \{0\}} \mu _{0}(j), \quad \mu _{0}(j) = \textrm{i}\,\beta {\mathtt L}(j) + {\mathtt z}_{0}(j) , \\&\quad \textbf{Z}_{0}:= {\mathcal {Z}}_{M}:= {{\,\textrm{diag}\,}}_{j\in {\mathbb {Z}}^2\setminus \{0\}} {\mathtt z}_{0}(j), \quad {\mathtt z}_{0} (j):= z_{M}(j) , \ \ j \in {\mathbb {Z}}^2\setminus \{0\}, \end{aligned} \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt L}(j)\in {\mathbb {R}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt z}_0 ( j )\in {\mathbb {C}}$$\end{document} defined respectively in (1.5) and in Proposition 5.6 (if we assume reversibility conditions, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt z}_{0}(j)\in \textrm{i}{\mathbb {R}}$$\end{document} );

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bullet $$\end{document} For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [s_0,S-\sigma _{M}]$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{0}:= {\mathcal {E}}_{M} \in {{\mathcal {O}}}{{\mathcal {P}}}{{\mathcal {M}}}_{s}^{- M}$$\end{document} satisfies the estimate (see (5.42))

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | \textbf{E}_{0} |_{-M,s}^{\textrm{Lip}(\gamma )} \lesssim _{s, M} \lambda ^{M(\theta -1)+1} \gamma ^{-(M-1)} \Vert w \Vert _{s+\sigma _{M}} \end{aligned}$$\end{document}

and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1 \geqslant s_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w_1, w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sigma _{0} \geqslant s_1 + \sigma _{M} $$\end{document} , then (see (5.43))

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\Delta _{12} \textbf{E}_0|_{- M, s_1} \lesssim _{s_1, M} \lambda ^{M(\theta -1)+1} \gamma ^{-(M-1)} \Vert w_1 - w_2 \Vert _{s_1 + \sigma _{M}}. \end{aligned}$$\end{document}

Note that, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \frac{1-{\mathtt c}}{2(1-{\mathtt c})-\alpha }$$\end{document} and assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\lambda ^{-{\mathtt c}}$$\end{document} , recalling also (4.5), (4.6), the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{0}$$\end{document} has small size with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough.

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau , {\mathtt N}_0 > 0$$\end{document} , we fix the constants

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&M:= \max \{2 \tau , \tfrac{1-{\mathtt c}}{2(1-{\mathtt c})-\alpha } \}\ + 1, \quad {\mathtt a}:= \textrm{max}\{3(2 \tau \! + \!M \!+ 1) + 1,\, \chi (\tau + \tau ^2 + 2) \} , \quad {\mathtt b}:=\! {\mathtt a}+ 1 , \\&\quad \tau _{1}:= 4\tau +2 +M , \quad \Sigma ({\mathtt b}):= \sigma _{M} +{\mathtt b}, \quad S > s_0 + \Sigma ({\mathtt b}), \\&{\mathtt N}_{- 1}:= 1 , \quad {\mathtt N}_n:= {\mathtt N}_0^{\chi ^n}, \quad n \geqslant 0, \quad \chi := 3/2 , \end{aligned} \end{aligned}$$\end{document}

where M, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{M }$$\end{document} are introduced in Proposition 5.6.

Remark 5.8

Let us describe of the parameters introduced in (5.56) and their role in the following Proposition 5.9. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt a}>0$$\end{document} appears in the negative exponent of the first estimate in (5.64) and measures how the low regularity norm of the remainder is fastly decreasing at each iteration. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt b}>0$$\end{document} appears in the regularity index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s+{\mathtt b}$$\end{document} of the second estimate in (5.64), which is the high regularity norm of the remainder that is allowed to diverge. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _1>0$$\end{document} appears as an exponent in the smallness condition (5.59). The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ({\mathtt b})>0$$\end{document} accounts for the loss of derivatives of the KAM iteration.

By Proposition 5.6, replacing s by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s + {\mathtt b}$$\end{document} in (5.42) and having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textbf{Z}_{0} = \textrm{diag}_{j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}} {\mathtt z}_{0}(j)$$\end{document} diagonal, and by the ansatz (4.2), one gets the initialization conditions for the KAM reducibility, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\sup _{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} |j| |{\mathtt z}_{0}(j)|^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta }, \\&\quad |\textbf{E}_{0}|_{- M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} \lesssim _{s} \lambda ^{M(\theta -1)+1} \gamma ^{-(M-1)} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )}, \\&\quad |\textbf{E}_{0}|_{- M, s_0 + {\mathtt b}}^{\textrm{Lip}(\gamma )} \lesssim _{s} \lambda ^{M(\theta -1)+1} \gamma ^{-(M-1)} , \\&\quad \text {and if} \quad w_1, w_2 \quad \text { satisfy } (4) \text { with} \quad \sigma _{0} = \Sigma ({\mathtt b}) \quad \text {then} \\&\quad |\Delta _{12} \textbf{E}_0 |_{- M, s_0 + \mathtt b} \lesssim \lambda ^{M(\theta -1)+1} \gamma ^{-(M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma (\mathtt b)}. \end{aligned} \end{aligned}$$\end{document}

By the definition of M in (5.56), one has that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M>\frac{1-{\mathtt c}}{2(1-{\mathtt c})-\alpha }> \frac{1}{1-\theta }$$\end{document} . We work in the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} and recalling (4.5), (4.6), we define the small parameters (recalling (5.2))

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon := \lambda ^{\theta -1}, \quad \varepsilon _{M}:= \lambda ^{M(\theta -1)+1} = \varepsilon ^{M-1} \lambda ^{\theta } . \end{aligned}$$\end{document}

Proposition 5.9

(Reducibility) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \Sigma ({\mathtt b})$$\end{document} , with the notation of (5.56). There exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_{0}:= {\mathtt N}_{0}(S, \tau ,\nu ) > 0$$\end{document} large enough and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau ,\nu ) \in (0, 1)$$\end{document} small enough such that, if (4.2) holds with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = \Sigma ({\mathtt b})$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt N}_{0}^{\tau _1} \varepsilon ^{M}\gamma ^{-M} \leqslant \delta , \end{aligned}$$\end{document}

then the following statements hold for any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} .

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S1)}_n$$\end{document} There exists a real, momentum preserving operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\textbf{L}_{n}:= \lambda \,\omega \cdot \partial _\varphi + \textbf{D}_{n} + \textbf{E}_{n}: H^{s + 1}_0 \rightarrow H^s_0 , \\&\quad \textbf{D}_{n}:= \beta \, {\mathtt L}+ \textbf{Z}_{n}= \textrm{diag}_{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} \mu _{n}(j) , \\&\quad \textbf{Z}_{n} = \textrm{diag}_{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} {\mathtt z}_{n}(j), \quad \mu _{n}(j):= \textrm{i}\, \beta {\mathtt L}(j)+ {\mathtt z}_{n}(j), \end{aligned} \end{aligned}$$\end{document}

defined for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n}^\gamma $$\end{document} , where we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{0}^\gamma := \Omega _{0}^\gamma (w):= {\mathtt D}{\mathtt C}(2\gamma , \tau )\cap \Lambda _{o}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} and, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Omega _{n}^\gamma :=\Omega _{n}^\gamma (w):= \Big \{ \omega \in \Omega _{n - 1}^\gamma : \, |\textrm{i}\,\lambda \,\omega \cdot \ell + \mu _{n - 1}(j) - \mu _{n - 1}(j') | \geqslant \frac{\lambda \, \gamma }{\langle \ell \rangle ^\tau |j'|^\tau }, \\&\qquad \ \forall \,\ell \in {\mathbb {Z}}^\nu \setminus \{ 0 \} , \ \ j,j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}, \ \ \pi ^\top ( \ell ) + j-j' =0, \ \ |\ell |\leqslant {\mathtt N}_{n-1} \Big \}. \end{aligned} \end{aligned}$$\end{document}

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} , the eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n}(j)= \mu _{n}(j;\omega )$$\end{document} satisfy the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&|{\mathtt z}_{n}(j)|^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta } |j|^{-1}\,, \quad | {\mathtt z}_{n}(j)- {\mathtt z}_{0}(j) |^{\textrm{Lip}(\gamma )} \lesssim \varepsilon _{M} \gamma ^{-(M-1)} |j|^{- M} \,, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&| {\mathtt z}_{n}(j) - {\mathtt z}_{n - 1}(j)|^{\textrm{Lip}(\gamma )} \lesssim \varepsilon _{M} \gamma ^{-(M-1)} \, {\mathtt N}_{n - 2}^{- {\mathtt a}}\, |j|^{- M} \quad \ \text {when } \quad n\geqslant 1 \,. \end{aligned}$$\end{document}

The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{n}$$\end{document} is real and momentum preserving, satisfying, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\textbf{E}_{n} |_{- M, s}^{\textrm{Lip}(\gamma )} \leqslant C_*(s) \varepsilon _{M} \gamma ^{-(M-1)} \, {\mathtt N}_{n - 1}^{- {\mathtt a}} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \\&\quad |\textbf{E}_{n} |_{-M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} \leqslant C_*(s) \varepsilon _{M} \gamma ^{-(M-1)} \, {\mathtt N}_{n - 1} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \end{aligned} \end{aligned}$$\end{document}

for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_* (s) > 0$$\end{document} large enough.

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , there exists an invertible, real, and momentum preserving map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{n -1} = \textrm{exp}(\Psi _{n - 1})$$\end{document} , such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n}^\gamma $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{L}_{n} = \Phi _{n - 1}^{- 1} \textbf{L}_{n-1} \Phi _{n - 1}. \end{aligned}$$\end{document}

Moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} , the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _{n - 1}: H^s_0 \rightarrow H^s_0$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Psi _{n - 1}|_{0, s}^{\textrm{Lip}(\gamma )}&\lesssim _s \varepsilon ^{M} \gamma ^{- M} {\mathtt N}_{n - 1}^{2 \tau + 1} {\mathtt N}_{n - 2}^{- {\mathtt a}} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )}, \\ |\Psi _{n - 1} |_{0, s + {\mathtt b}}^{\textrm{Lip}(\gamma )}&\lesssim _s \varepsilon ^{M} \gamma ^{- M} {\mathtt N}_{n - 1}^{2 \tau + 1} {\mathtt N}_{n - 2} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

In addition, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_{n}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{n}$$\end{document} are reversible, the eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n}(j)= \mu _{n}(j;\omega )$$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} , are purely imaginary, satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\mu _n(j) = - \mu _n(- j) = - \overline{\mu _n( j)} , \ \ \text {or equivalently} \\&\quad {\mathtt z}_{n}(j) = - {\mathtt z}_{n}(- j)= - \overline{{\mathtt z}_{n}( j)}, \end{aligned} \end{aligned}$$\end{document}

and, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 1$$\end{document} , the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{n - 1}$$\end{document} is reversibility preserving.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S2)}_n$$\end{document} For all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ j \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} , there exists a Lipschitz extension of the eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _n(j;\,\cdot \,):\Omega _{n}^\gamma \rightarrow {\mathbb {C}}$$\end{document} to the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma , \tau )\cap \Lambda _{o}$$\end{document} , denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{\mu }_n(j;\,\cdot \,): {{\mathtt D}{\mathtt C}(2\gamma , \tau ) \cap \Lambda _{o}}\rightarrow {\mathbb {C}}$$\end{document} , satisfying, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{{\widetilde{\mu }}}_n(j) - {{\widetilde{\mu }}}_{n - 1}(j) |^{\textrm{Lip}(\gamma )} \lesssim |j|^{- M} |\textbf{E}_{n - 1}|_{-M,s_0}^{\textrm{Lip}(\gamma )} \lesssim \varepsilon _{M} \gamma ^{-(M-1)} \, {\mathtt N}_{n - 2}^{- {\mathtt a}}\, |j|^{- M}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S3)_{n}}$$\end{document} Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1(\omega )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w_2(\omega ) $$\end{document} satisfying the ansatz (4.2) be such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_0(w_1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_0(w_2 )$$\end{document} satisfy (5.57). Then for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _n^{\gamma _1}(w_1) \cap \Omega _n^{\gamma _2}(w_2)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1, \gamma _2 \in [\gamma /2, 2 \gamma ]$$\end{document} , the following estimates hold:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Delta _{12} \textbf{E}_n |_{- M, s_0} \leqslant C_*(s) {\mathtt N}_{n - 1}^{- {\mathtt a}} \Vert w_1 - w_2\Vert _{s_0 + \Sigma ({\mathtt b})} , \\&\quad |\Delta _{12} \textbf{E}_n |_{- M, s_0 + {\mathtt b}} \leqslant C_*(s) {\mathtt N}_{n - 1} \Vert w_1 - w_2\Vert _{s_0 + \Sigma ({\mathtt b})} . \end{aligned} \end{aligned}$$\end{document}

Moreover for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \big |\Delta _{12}(\mu _n(j) - \mu _{n - 1}(j)) \big |&= \big |\Delta _{12}({\mathtt z}_n(j) - {\mathtt z}_{n - 1}(j)) \big | \\&\lesssim \varepsilon _{M} \gamma ^{-(M-1)} \, {\mathtt N}_{n - 2}^{- {\mathtt a}} \, |j|^{- M} \Vert w_1 - w_2 \Vert _{ s_0 + \Sigma ({\mathtt b})}, \\ |\Delta _{12} \mu _n(j)| = |\Delta _{12} {\mathtt z}_{n}(j)|&\lesssim \lambda ^{\theta } |j|^{- 1} \Vert w_1 - w_2 \Vert _{ s_0 + \Sigma ({\mathtt b})}. \end{aligned} \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)_{n}}$$\end{document} Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_2$$\end{document} be like in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S3)_{n}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \rho \leqslant \gamma /2$$\end{document} . Then:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K \varepsilon \rho ^{- 1} {\mathtt N}_{n - 1}^{\tau + \tau ^2} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} \leqslant 1 \quad \Longrightarrow \quad \Omega _n^\gamma (w_1) \subseteq \Omega _n^{\gamma - \rho }(w_2) \,. \end{aligned}$$\end{document}

Proof

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S1)}_0-\mathbf{(S4)}_0$$\end{document} . The claimed properties follow directly from Proposition 5.6, recalling (5.55), (5.57), (5.58) and the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{0}^\gamma := {\mathtt D}{\mathtt C}(2\gamma , \tau )\cap \Lambda _{o}$$\end{document} .

By induction, we assume the the claimed properties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S1)}_n$$\end{document} - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)}_n$$\end{document} hold for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} and we prove them at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} .

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S1)}_{n+1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _n = \textrm{exp}( \Psi _n)$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} is an operator of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- M$$\end{document} to be determined. By the Lie expansion, we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \textbf{L}_{n + 1}&= \Phi _n^{- 1}\textbf{L}_{n} \Phi _n = \lambda \, \omega \cdot \partial _\varphi + \textbf{D}_n + \lambda \, \omega \cdot \partial _\varphi \Psi _n + [\textbf{D}_n, \Psi _n] + \Pi _{{\mathtt N}_n} \textbf{E}_n + \textbf{E}_{n + 1} , \\ \textbf{E}_{n + 1}&:= \Pi _{{\mathtt N}_n}^\bot \textbf{E}_n + \int _0^1 (1 - \tau ) \textrm{exp}(- \tau \Psi _n) \,[\lambda \,\omega \cdot \partial _\varphi \Psi _n + [\textbf{D}_n, \Psi _n], \Psi _n] \, \textrm{exp}(\tau \Psi _n) \,\textrm{d}{\tau }\\&\quad + \int _0^1 \textrm{exp}(- \tau \Psi _n) [\textbf{E}_n, \Psi _n] \textrm{exp}(\tau \Psi _n) \,\textrm{d}{\tau }, \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_{n}:= \beta \,{\mathtt L}+ \textbf{Z}_{n}$$\end{document} and the projectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{N}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{N}^\bot $$\end{document} are defined in (2.16). Our purpose is to find a map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} solving the homological equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda \,\omega \cdot \partial _\varphi \Psi _n + [\textbf{D}_{n}, \Psi _n] + \Pi _{{\mathtt N}_{n}} \textbf{E}_{n} = {\widehat{\textbf{E}}}_n(0) , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\textbf{E}}}_n(0):= \frac{1}{(2 \pi )^\nu } \int _{{\mathbb {T}}^\nu } \textbf{E}_n(\varphi ) \,\textrm{d}{\varphi }$$\end{document} is a diagonal operator by Lemma 2.13, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{n}({\varphi })$$\end{document} is a momentum preserving operator by induction assumption.

By (2.9) and (5.60), the homological equation (5.73) is equivalent to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\big ( \textrm{i}\,\lambda \, \,\omega \cdot \ell + \mu _{n}(j) - \mu _{n}(j') \big ) {{\widehat{\Psi }}}_{n}(\ell )_j^{j'} + \widehat{\textbf{E}_{n}}(\ell )_j^{j'} = 0, \\&\quad \ell \in {\mathbb {Z}}^\nu \setminus \{0\}, \ \ |\ell | \leqslant {\mathtt N}_n,\, \ \ j, j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}, \ \ \pi ^\top (\ell ) + j - j' = 0. \end{aligned} \end{aligned}$$\end{document}

Therefore, we define the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _{n}$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\widehat{\Psi }}}_{n} (\ell )_j^{j'}:= {\left\{ \begin{array}{ll} - \dfrac{\widehat{\textbf{E}_{n}}(\ell )_j^{j'} }{ \textrm{i}\,\lambda \, \omega \cdot \ell + \mu _{n}(j) - \mu _{n}(j') }, & \begin{matrix} \ell \in {\mathbb {Z}}^{\nu }\setminus \{0\} , \quad j,j'\in {\mathbb {Z}}^2\setminus \{0\}, \\ |\ell | \leqslant {\mathtt N}_{n}, \quad \pi ^\top (\ell ) + j-j'=0, \end{matrix} \\ 0 & \text {otherwise}. \end{array}\right. } \end{aligned}$$\end{document}

which is a solution of (5.73)-(5.74).

Lemma 5.10

The operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} in (5.75), defined for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^\gamma $$\end{document} , satisfies, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Psi _n|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s} {\mathtt N}_{n}^{2 \tau + 1} \lambda ^{- 1} \gamma ^{- 1} |\textbf{E}_{n}|_{- M, s}^{\textrm{Lip}(\gamma )}, \\&\quad |\Psi _n|_{0, s + \eta }^{\textrm{Lip}(\gamma )} \lesssim _{s} {\mathtt N}_{n}^{2 \tau + 1 + \eta } \lambda ^{- 1}\gamma ^{- 1} |\textbf{E}_{n}|_{- M, s}^{\textrm{Lip}(\gamma )}, \quad \forall \, \eta > 0 . \end{aligned} \end{aligned}$$\end{document}

Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1,\, w_2$$\end{document} satisfy (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = \Sigma ({\mathtt b})$$\end{document} , as in (5.56). Then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^{\gamma _1}(w_1) \cap \Omega _{n + 1}^{\gamma _2}(w_2)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /2 \leqslant \gamma _1, \gamma _2 \leqslant \gamma $$\end{document} , one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Delta _{12} \Psi _n|_{0, s_0}&\lesssim \varepsilon ^{M} \gamma ^{-M} {\mathtt N}_{n}^{2 \tau } {\mathtt N}_{n - 1}^{- {\mathtt a}} \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} , \\ |\Delta _{12} \Psi _n |_{0, s_0 + {\mathtt b}}&\lesssim \varepsilon ^{M} \gamma ^{-M} {\mathtt N}_{n}^{2 \tau } {\mathtt N}_{n - 1}\Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} , \\ |\Delta _{12} \Psi _n|_{0, s_0 + \eta }&\lesssim \varepsilon ^{M} \gamma ^{-M} {\mathtt N}_{n}^{2 \tau + \eta } {\mathtt N}_{n - 1}^{- {\mathtt a}} \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} , \quad \forall \, \eta> 0 ,\\ |\Delta _{12} \Psi _n |_{0, s_0 + {\mathtt b}+ \eta }&\lesssim \varepsilon ^{M} \gamma ^{-M} {\mathtt N}_{n}^{2 \tau + \eta } {\mathtt N}_{n - 1}\Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} , \quad \forall \, \eta > 0. \end{aligned} \end{aligned}$$\end{document}

Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} is real and momentum preserving. In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} is also reversibility preserving.

Proof

To simplify notations, along this proof we drop the index n.

Proof of (5.76).

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\Psi }}(\ell )_{j}^{j'}={\widehat{\Psi }}(\ell ;\omega )_{j}^{j'}$$\end{document} as in (5.75), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ell \in {\mathbb {Z}}^\nu $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ j, j' \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < |\ell | \leqslant {\mathtt N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi ^\top (\ell ) + j - j' = 0$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^\gamma $$\end{document} (see (5.61)), we immediately get the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{{\widehat{\Psi }}} (\ell ;\omega )_j^{j'}| \lesssim \lambda ^{- 1}\gamma ^{- 1} \langle \ell \rangle ^\tau | j' |^\tau |{\widehat{\textbf{E}}}(\ell ;\omega )_j^{j'}| \lesssim \lambda ^{- 1} \gamma ^{- 1} {\mathtt N}^\tau | j' |^\tau |{\widehat{\textbf{E}}}(\ell ;\omega )_j^{j'}| . \end{aligned}$$\end{document}

We define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{\ell j j'}(\omega ):=\textrm{i}\,\lambda \, \omega \cdot \ell +\mu (j;\omega ) - \mu (j';\omega )$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{1}, \omega _{2} \in \Omega _{n + 1}^\gamma $$\end{document} . By (5.60), (5.61), (5.62), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\big | \big ( \mu (j;\omega _1) - \mu (j';\omega _1) \big ) - \big ( \mu (j;\omega _2) - \mu (j';\omega _2)\big ) \big | \\&\quad \leqslant \big |{\mathtt z}(j; \omega _1) - {\mathtt z}(j; \omega _2)\big | + \big |{\mathtt z}(j'; \omega _1) - {\mathtt z}(j'; \omega _2)\big | \\&\quad \lesssim \lambda ^{\theta } \gamma ^{- 1} \big ( |j|^{-1} + |j'|^{-1} \big ) |\omega _1 - \omega _2| \lesssim \lambda ^{\theta } \gamma ^{- 1} |\omega _1 - \omega _2| , \end{aligned} \end{aligned}$$\end{document}

and therefore, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1}\gamma ^{-1}\ll 1 $$\end{document} by (4.7),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\delta _{\ell j j'}(\omega _1) - \delta _{\ell j j'}(\omega _2)|&\lesssim (\lambda \, |\ell |+\lambda ^{\theta }\gamma ^{-1}) |\omega _1 - \omega _2| \lesssim \lambda \, \langle \ell \rangle |\omega _1 - \omega _2|. \end{aligned} \end{aligned}$$\end{document}

Then, estimate (5.79), together with the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1, \omega _2 \in \Omega _{n + 1}^\gamma $$\end{document} , implies that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Big | \frac{1}{ \delta _{\ell j j'}(\omega _{1})} - \frac{1}{\delta _{\ell j j'}(\omega _{2})} \Big |&\leqslant \dfrac{|\delta _{\ell j j'}(\omega _{1}) - \delta _{\ell j j'}(\omega _{2})|}{|\delta _{\ell j j'}(\omega _{1})| |\delta _{\ell j j'}(\omega _{2})|} \\&\lesssim \lambda (\lambda \gamma )^{-2} \langle \ell \rangle ^{2 \tau + 1} |j'|^{2 \tau } |\omega _{1} - \omega _{2} | \\&\lesssim \lambda ^{- 1} \gamma ^{- 2} \langle \ell \rangle ^{2 \tau + 1} |j'|^{2 \tau } |\omega _{1} - \omega _{2} |. \end{aligned} \end{aligned}$$\end{document}

Therefore, by (5.75) and (5.80), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{1},\omega _{2}\in \Omega _{n + 1}^\gamma $$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \big |{{\widehat{\Psi }}} (\ell ;\omega _{1})_j^{j'} - {{\widehat{\Psi }}} (\ell ;\omega _{2})_j^{j'}\big |&\lesssim \langle \ell \rangle ^\tau |j'|^\tau \lambda ^{- 1} \gamma ^{- 1} \big |{{\widehat{\textbf{E}}}}(\ell ;\omega _{1})_j^{j'} - {{\widehat{\textbf{E}}}}(\ell ;\omega _{2})_j^{j'}\big | \\&\ \ + \langle \ell \rangle ^{2 \tau + 1} |j'|^{2 \tau } \lambda ^{- 1} \gamma ^{-2} \big |{\widehat{\textbf{E}}}(\ell ;\omega _{2})_j^{j'}\big | |\omega _{1}-\omega _{2} | \\&\lesssim {\mathtt N}^\tau |j'|^\tau \lambda ^{- 1} \gamma ^{- 1} \big |{{\widehat{\textbf{E}}}}(\ell ;\omega _{1})_j^{j'} - {{\widehat{\textbf{E}}}}(\ell ;\omega _{2})_j^{j'}\big | \\&\ \ + {\mathtt N}^{2 \tau + 1} |j'|^{2 \tau } \lambda ^{- 1} \gamma ^{-2} \big |{\widehat{\textbf{E}}}(\ell ;\omega _{2})_j^{j'}\big | |\omega _{1}-\omega _{2} |. \end{aligned} \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > 2 \tau $$\end{document} by (5.56), recalling Definition 2.5 and collecting the estimates (5.78), (5.81), we obtain the bounds, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [s_0,S+\Sigma ({\mathtt b})]$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Psi |_{0, s}^\textrm{sup} \lesssim {\mathtt N}^{ \tau } \lambda ^{- 1}\gamma ^{- 1} |{\textbf{E}}|_{- M, s}^\textrm{sup}, \\&\quad |\Psi |_{0, s}^\textrm{lip} \lesssim {\mathtt N}^{2 \tau + 1} \lambda ^{- 1}\gamma ^{- 2} |{\textbf{E}}|_{- M, s}^\textrm{sup} + {\mathtt N}^{\tau } \lambda ^{- 1} \gamma ^{- 1} |{\textbf{E}}|_{- M, s}^\textrm{lip}. \end{aligned} \end{aligned}$$\end{document}

Similarly, using also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} imply that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|j - j'| \lesssim |\ell | \lesssim {\mathtt N}$$\end{document} , by analogous arguments we obtain that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta > 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Psi |_{0, s + \eta }^\textrm{sup} \lesssim {\mathtt N}^{\tau + \eta } \gamma ^{- 1} \lambda ^{- 1} |{\textbf{E}}|_{- M, s}^\textrm{sup}, \\&\quad |\Psi |_{0, s+ \eta }^\textrm{lip} \lesssim {\mathtt N}^{2 \tau + \eta +1} \gamma ^{- 2} \lambda ^{- 1} |{\textbf{E}}|_{- M, s}^\textrm{sup} + {\mathtt N}^{\tau + \eta } \lambda ^{- 1} \gamma ^{- 2} |{\textbf{E}}|_{- M, s}^\textrm{sup}. \end{aligned} \end{aligned}$$\end{document}

Hence, we conclude the claimed bounds in (5.76).

Proof of (5.77). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^{\nu }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^2 {\setminus } \{ 0\}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < |\ell | \leqslant {\mathtt N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} . Recalling (5.60), we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\delta _{\ell j j'}(w_i):= \textrm{i}\, \lambda \,\omega \cdot \ell + \mu (j; w_i) - \mu (j'; w_i) , \quad i = 1,2, \\&\quad \mu (j; w_i):= \textrm{i}\, \beta \,{\mathtt L}(j)+ {\mathtt z}(j; w_i). \end{aligned} \end{aligned}$$\end{document}

By (5.70), we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Delta _{12} \delta _{\ell j j'}|&\leqslant |\Delta _{12} {\mathtt z}(j)| + |\Delta _{12} {\mathtt z}(j')| \lesssim \lambda ^{\theta }\big (|j|^{- 1} + |j'|^{- 1} \big ) \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} \\&\lesssim \lambda ^{\theta } \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} . \end{aligned} \end{aligned}$$\end{document}

Therefore, by (5.75), (5.96) and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1}\gamma ^{-1}\ll 1$$\end{document} , we get, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^{\gamma _1}(w_1) \cap \Omega _{n + 1}^{\gamma _1}(w_2)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /2 \leqslant \gamma _1, \gamma _2 \leqslant \gamma $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \big |\Delta _{12} {\widehat{\Psi }}(\ell )_j^{j'}\big |&\leqslant \frac{\big |\Delta _{12} {{\widehat{\textbf{E}}}}(\ell )_j^{j'}\big |}{|\delta _{\ell j j'}(w_1)|} + \frac{\big |{{\widehat{\textbf{E}}}}(\ell )_j^{j'}(w_2\big )| |\Delta _{12} \delta _{\ell j j'}|}{|\delta _{\ell j j'}(w_1)| |\delta _{\ell j j'}(w_2)|} \\&\lesssim {\mathtt N}^\tau \lambda ^{- 1} \gamma ^{- 1} |j'|^\tau \big |\Delta _{12} {{\widehat{\textbf{E}}}}(\ell )_j^{j'}\big | \\&\quad + {\mathtt N}^{2 \tau } \lambda ^{\theta } \lambda ^{- 2} \gamma ^{- 2} |j'|^{2 \tau }\big |{{\widehat{\textbf{E}}}}(\ell )_j^{j'}(w_2)\big | \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})} \\&\lesssim {\mathtt N}^{2 \tau } \lambda ^{- 1} \gamma ^{- 1} |j'|^{2 \tau } \big ( \big |\Delta _{12} {{\widehat{\textbf{E}}}}(\ell )_j^{j'}\big | + \big |{{\widehat{\textbf{E}}}}(\ell )_j^{j'}(w_2)\big | \Vert w_1 - w_2 \Vert _{s_1 + \Sigma ({\mathtt b})}\big ) . \end{aligned} \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > 2 \tau $$\end{document} by (5.56), recalling Definition 2.5 and the induction estimates (5.64), (5.69), the claimed bounds follow easily. Finally, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{E}}$$\end{document} is real, reversible and momentum preserving, by (5.75), Lemma 2.9 and Lemma 2.13 we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} is real, reversibility preserving and momentum preserving. This concludes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

By Lemma 5.10, the induction assumption on the estimates (5.64) and by (5.58), we obtain, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Psi _n|_{0, s}^{\textrm{Lip}(\gamma )}&\lesssim _{s} {\mathtt N}_n^{2 \tau + 1} \lambda ^{- 1}\gamma ^{- 1} |{\textbf{E}}_n |_{- M, s}^{\textrm{Lip}(\gamma )} \lesssim _{s } {\mathtt N}_n^{2 \tau + 1} {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^M \gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \\ |\Psi _n|_{0, s + {\mathtt b}}^{\textrm{Lip}(\gamma )}&\lesssim _{s} {\mathtt N}_n^{2 \tau + 1} \lambda ^{- 1} \gamma ^{- 1} |{\textbf{E}}_n |_{-M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} \lesssim _{s } {\mathtt N}_n^{2 \tau + 1} {\mathtt N}_{n - 1} \varepsilon ^{M}\gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )}, \end{aligned} \end{aligned}$$\end{document}

which are the estimates (5.66) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . Moreover, setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =M$$\end{document} in (5.76), by the same arguments we also have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Psi _n|_{0, s + M}^{\textrm{Lip}(\gamma )} ,\,&\lesssim _{s} {\mathtt N}_n^{2 \tau + 1 + M} \gamma ^{- 1} \lambda ^{- 1} |{\textbf{E}}_n |_{- M, s}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s } {\mathtt N}_n^{2 \tau + 1 + M} {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^M \gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \\ |\Psi _n|_{0, s + {\mathtt b}+ M}^{\textrm{Lip}(\gamma )}&\lesssim _{s} {\mathtt N}_n^{2 \tau + 1 + M} \lambda ^{- 1}\gamma ^{- 1} |{\textbf{E}}_n |_{-M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} \\&\lesssim _{s } {\mathtt N}_n^{2 \tau + M + 1} {\mathtt N}_{n - 1} \varepsilon ^M \gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

In particular, by (5.56), (4.2) and by the smallness condition (5.59), we deduce, setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=s_0$$\end{document} in (5.83), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_0>0$$\end{document} large enough,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Psi _n|_{0, s_0 + M}^{\textrm{Lip}(\gamma )}&\lesssim {\mathtt N}_n^{2 \tau +1 + M} {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^M \gamma ^{- M} \Vert w \Vert _{s_0 + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \ \leqslant \delta < 1. \end{aligned} \end{aligned}$$\end{document}

Therefore, by Lemma 2.6-(iv) and estimates (5.83), (5.84), we have the estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\sup _{\tau \in [- 1, 1]}|\textrm{exp}(\tau \Psi _n)|_{0, s + M}^{\textrm{Lip}(\gamma )} \lesssim _s 1 + |\Psi _n|_{0, s + M}^{\textrm{Lip}(\gamma )}, \quad \sup _{\tau \in [- 1, 1]}|\textrm{exp}(\tau \Psi _n)|_{0, s_0 + M}^{\textrm{Lip}(\gamma )} \lesssim 1, \\&\quad \sup _{\tau \in [- 1, 1]}|\textrm{exp}(\tau \Psi _n)|_{0, s + {\mathtt b}+ M}^{\textrm{Lip}(\gamma )} \lesssim _s 1 + |\Psi _n|_{0, s + {\mathtt b}+ M}^{\textrm{Lip}(\gamma )}. \end{aligned} \end{aligned}$$\end{document}

Then, by recalling (5.72) and by using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n$$\end{document} solves the equation (5.73), we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\textbf{L}}_{n + 1}&\,=\lambda \, \omega \cdot \partial _\varphi + {\textbf{D}}_{n + 1} + {\textbf{E}}_{n + 1}, \\ {\textbf{D}}_{n + 1}&:= -\beta {\mathtt L}+ {\textbf{Z}}_{n + 1}, \qquad {\textbf{Z}}_{n + 1}: = {\textbf{Z}}_n + {\widehat{\textbf{E}}}_n(0), \\ {\textbf{E}}_{n + 1}&\,=\Pi _{{\mathtt N}_n}^\bot \textbf{E}_n + \int _0^1 (1 - \tau ) \textrm{exp}(- \tau \Psi _n) \,[{\widehat{\textbf{E}}}_n(0) - \Pi _{{\mathtt N}_n} \textbf{E}_n, \Psi _n] \, \textrm{exp}(\tau \Psi _n)\,\textrm{d}{\tau }\\&\quad + \int _0^1 \textrm{exp}(- \tau \Psi _n) [\textbf{E}_n, \Psi _n] \textrm{exp}(\tau \Psi _n) \,\textrm{d}{\tau }. \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

All the operators in (5.86)

are defined for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^\gamma $$\end{document} . Moreover, by (5.72), (5.73), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega ^\gamma _{n+1}$$\end{document} one has the identity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _n^{-1} \textbf{L}_n \Phi _n = \textbf{L}_{n+1}$$\end{document} , which is (5.65) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . Recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\textbf{E}}}_{n}(0)$$\end{document} is a diagonal operator and by (5.86), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {\textbf{Z}}_{n + 1}&= \textrm{diag}_{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} {\mathtt z}_{n + 1}(j), \quad {\mathtt z}_{n + 1}(j):= {\mathtt z}_n(j) + {\widehat{\textbf{E}}}_n(0)_j^j, \\ {\textbf{D}}_{n + 1}&= \textrm{diag}_{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} \mu _{n + 1}(j), \quad \mu _{n + 1}(j):= \textrm{i}\, \beta \, {\mathtt L}(j)+ {\mathtt z}_{n + 1}(j). \end{aligned} \end{aligned}$$\end{document}

By Lemma 2.6-(v)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} | \mu _{n + 1}(j) - \mu _n(j)|^{\textrm{Lip}(\gamma )}&= | {\mathtt z}_{n + 1}(j) - {\mathtt z}_n(j)|^{\textrm{Lip}(\gamma )} \\&\leqslant | {\widehat{\textbf{E}}}_n(0)_j^j |^{\textrm{Lip}(\gamma )} \lesssim |{\textbf{E}}_n|_{- M, s_0}^{\textrm{Lip}(\gamma )} \langle j \rangle ^{- M}. \end{aligned} \end{aligned}$$\end{document}

Then, (5.88), together with the estimate (5.64),

implies (5.63) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . The estimate (5.62) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} follows, as usual, by a telescoping argument, using the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n \geqslant 0} {\mathtt N}_{n - 1}^{- {\mathtt a}}$$\end{document} is convergent since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt a}> 0$$\end{document} (see (5.56)). Now we prove the estimates (5.64) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . By (5.86), estimates (5.82), (5.84), (5.85), the induction estimates (5.64), Lemma 2.6-(ii), (v) and Lemma 2.7, we get, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s \leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|{\textbf{E}}_{n + 1}|_{- M, s}^{\textrm{Lip}(\gamma )} \lesssim _{s} {\mathtt N}_n^{- {\mathtt b}} |{\textbf{E}}_n|_{- M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} + {\mathtt N}_n^{2 \tau + 1 + M}\lambda ^{- 1} \gamma ^{- 1} |{\textbf{E}}_n|_{- M, s_0}^{\textrm{Lip}(\gamma )} |{\textbf{E}}_n|_{- M, s}^{\textrm{Lip}(\gamma )} , \\&\quad |{\textbf{E}}_{n + 1}|_{- M, s + {\mathtt b}}^{\textrm{Lip}(\gamma )} \lesssim _{s} |{\textbf{E}}_n|_{- M, s +{\mathtt b}}^{\textrm{Lip}(\gamma )} \\&\qquad + {\mathtt N}_{n}^{2 \tau + 1 + M} \lambda ^{- 1}\gamma ^{-1}\big ( |{\textbf{E}}_n|_{- M, s_0}^{\textrm{Lip}(\gamma )} |{\textbf{E}}_n|_{- M, s+{\mathtt b}}^{\textrm{Lip}(\gamma )} + |{\textbf{E}}_n|_{- M, s}^{\textrm{Lip}(\gamma )} |{\textbf{E}}_n|_{- M, s_0+{\mathtt b}}^{\textrm{Lip}(\gamma )} \big ) . \end{aligned} \end{aligned}$$\end{document}

We now verify the estimates (5.64) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . By (5.89), the induction assumption on the estimate (5.64), (5.58) and using the ansatz (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = \Sigma ({\mathtt b})$$\end{document} , we get, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S - \Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |{\textbf{E}}_{n + 1}|_{- M, s +{\mathtt b}}^{\textrm{Lip}(\gamma )}&\leqslant C(s) C_*(s){\mathtt N}_{n - 1} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w\Vert _{s+\Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \\&+ 2C(s)C_*^2(s) C_0 \,{\mathtt N}_n^{2 \tau + M + 1} {\mathtt N}_{n - 1}^{- {\mathtt a}}\varepsilon ^{M} \gamma ^{- M} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \\&\leqslant C_*(s) {\mathtt N}_n \varepsilon _{M} \gamma ^{-(M-1)} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

provided

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(s) {\mathtt N}_{n}^{-1} {\mathtt N}_{n - 1} \leqslant \frac{1}{2} , \quad 2 C(s)C_*(s) C_0\, {\mathtt N}_n^{2 \tau + M } {\mathtt N}_{n - 1}^{ - {\mathtt a}}\varepsilon ^{M} \gamma ^{- M} \leqslant \frac{1}{2}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |{\textbf{E}}_{n + 1}|_{- M, s}^{\textrm{Lip}(\gamma )}&\leqslant C(s) C_*(s) {\mathtt N}_n^{- {\mathtt b}} {\mathtt N}_{n - 1} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w \Vert _{s+ \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \\&+ C(s) C_*^2(s) C_0\, {\mathtt N}_n^{2 \tau + M + 1} {\mathtt N}_{n - 1}^{- 2 {\mathtt a}} \varepsilon ^{-M} \gamma ^{-M} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w \Vert _{s+ \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \\&\leqslant C_*(s) {\mathtt N}_n^{- {\mathtt a}} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \end{aligned} \end{aligned}$$\end{document}

provided

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(s) {\mathtt N}_n^{{\mathtt a}- {\mathtt b}} {\mathtt N}_{n - 1} \leqslant \frac{1}{2} , \quad C(s) C_*(s)C_0 \, {\mathtt N}_n^{2 \tau + {\mathtt a}+ M + 1} {\mathtt N}_{n - 1}^{- 2 {\mathtt a}}\varepsilon ^{M} \gamma ^{- M} \leqslant \frac{1}{2} ; \end{aligned}$$\end{document}

note that the constant C(s) comes from estimates (5.89), the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_*(s)$$\end{document} comes from the estimates (5.64) and the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document} comes from impositing the ansatz (4.2). The conditions (5.91), (5.93) are verified by (5.56), (5.59), taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_0, \lambda \gg 0$$\end{document} large enough. Therefore, the estimates (5.90) and (5.92) imply that the estimates (5.64) hold at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . Finally, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n, \Phi _n, \Phi _n^{- 1}$$\end{document} are real and momentum preserving by Lemma 5.73 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{D}}_n, {\textbf{E}}_n$$\end{document} are real and momentum preserving by the induction assumption, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{D}}_{n + 1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{E}}_{n + 1}$$\end{document} are real and momentum preserving operators, by (5.86) and Lemmata 2.9, 2.11, 2.13. Morover if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _n, \Phi _n, \Phi _n^{- 1}$$\end{document} are reversibility preserving by Lemma 5.73 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{D}}_n, {\textbf{E}}_n$$\end{document} are reversible by the induction assumption, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{D}}_{n + 1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{E}}_{n + 1}$$\end{document} are reversible by (5.86) and Lemma 2.9 and hence (5.67) is verified at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} .

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S2)}_{n + 1}$$\end{document} . We now construct a Lipschitz extension for the eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n + 1}(j,\,\cdot \,): \Omega _{n + 1}^\gamma \rightarrow \,{\mathbb {C}}$$\end{document} . By the induction hypothesis, there exists a Lipschitz extension of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _n(j;\omega )$$\end{document} , denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widetilde{\mu }}}_{n}(j;\omega )$$\end{document} , to the whole set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma , \tau )$$\end{document} that satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S2)}_n$$\end{document} . By (5.87), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n + 1}(j) = \mu _n(j) + {\mathtt r}_n(j)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt r}_n(j, \omega )={\mathtt r}_n(j,\omega ):= {\widehat{\textbf{E}}}_n(0;\omega )_j^j$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathtt r}_n(j)|^{\textrm{Lip}(\gamma )} \lesssim {\mathtt N}_{n - 1}^{- {\mathtt a}} |j|^{- M} $$\end{document} .

Hence, by the Kirszbraun Theorem (see Lemma M.5 [39]) there exists a Lipschitz extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widetilde{{\mathtt r}}}}_n(j,\,\cdot \,): {{\mathtt D}{\mathtt C}(2\gamma , \tau ) \cap \Lambda _{o}}\rightarrow {\mathbb {C}}$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt r}_n(j,\,\cdot \,): \Omega _{n + 1}^\gamma \rightarrow {\mathbb {R}}$$\end{document} satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\widetilde{{\mathtt r}}}}_n(j)|^{\textrm{Lip}(\gamma )} \lesssim |{\mathtt r}_n(j)|^{\textrm{Lip}(\gamma )} \lesssim {\mathtt N}_{n - 1}^{- {\mathtt a}} |j|^{- M}$$\end{document} . The claimed statement then follows by defining \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widetilde{\mu }}}_{n + 1}(j):= {{\widetilde{\mu }}}_n(j) + {{\widetilde{{\mathtt r}}}}_n(j)$$\end{document} .

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S3)}_{n + 1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1, w_2$$\end{document} satisfy the ansatz (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = \Sigma ({\mathtt b})$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^{\gamma _1}(w_1) \cap \Omega _{n + 1}^{\gamma _2}(w_2)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1, \gamma _2 \in [\gamma /2\,,\, 2 \gamma ]$$\end{document} . Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _{12}{{\mathcal {F}}}_\tau := {{\mathcal {F}}}_\tau (w_1) - {\mathcal F}_\tau (w_2) , \quad {{\mathcal {F}}}_\tau := \textrm{exp}(\tau \Psi _n), \quad \tau \in [-1, 1]. \end{aligned}$$\end{document}

By Lemmata 2.6-(iv), 5.10 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1=s_0$$\end{document} , together with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta _{12} {\mathcal {F}}_{\tau } = (\textrm{exp } (\tau \Delta _{12} \Psi _{n} )-\textrm{Id}) {\mathcal {F}}_{\tau } (w_2)$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Delta _{12} {{\mathcal {F}}}_\tau |_{0, s_0} \lesssim {\mathtt N}_{n}^{2 \tau } {\mathtt N}_{n - 1}^{- {\mathtt a}}\varepsilon ^{M} \gamma ^{-M} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \\&\quad |\Delta _{12} {{\mathcal {F}}}_\tau |_{0, s_0 + M} \lesssim {\mathtt N}_{n}^{2 \tau + M} {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^{M} \gamma ^{-M}\Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \\&\quad |\Delta _{12} {{\mathcal {F}}}_\tau |_{0, s_0 + \beta } \lesssim {\mathtt N}_{n}^{2 \tau } {\mathtt N}_{n - 1} \varepsilon ^{M} \gamma ^{-M}\Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} , \\&\quad |\Delta _{12} {{\mathcal {F}}}_\tau |_{0, s_0 + M + \beta } \lesssim {\mathtt N}_{n}^{2 \tau + M} {\mathtt N}_{n - 1} \varepsilon ^{M} \gamma ^{-M}\Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}. \end{aligned} \end{aligned}$$\end{document}

We now shortly describe how to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12} \textbf{E}_{n + 1}$$\end{document} (recall the expression given in (5.86)). By Lemma 2.7 and the estimates in (5.69), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Delta _{12} \Pi _{{\mathtt N}_n}^\bot \textbf{E}_n|_{- M, s_0}&\lesssim {\mathtt N}_n^{- {\mathtt b}} |\Delta _{12} \textbf{E}_n|_{- M, s_0 + {\mathtt b}} \lesssim {\mathtt N}_n^{- {\mathtt b}} {\mathtt N}_{n - 1} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \\ |\Delta _{12} \Pi _{{\mathtt N}_n}^\bot \textbf{E}_n|_{- M, s_0+ {\mathtt b}}&\leqslant |\Delta _{12} \textbf{E}_n|_{- M, s_0+ {\mathtt b}} \lesssim {\mathtt N}_{n - 1} \Vert w_1 - w_2\Vert _{s_0 + \Sigma ({\mathtt b})}. \end{aligned} \end{aligned}$$\end{document}

All the terms appearing in the two integrals in the formula of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{E}_{n + 1}$$\end{document} (see (5.86)) can be all estimated in a similar way. Hence, for sake of simplicity in the exposition, we concentrate on the estimate of the term

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{B}:= \int _0^1 \textbf{V}_{\tau } \,\textrm{d}{\tau }, \quad \textbf{V}_{\tau }:= {{\mathcal {F}}}_{- \tau } \textbf{E}_n \Psi _n {{\mathcal {F}}}_\tau . \end{aligned}$$\end{document}

Clearly it is enough to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{V}_{\tau }$$\end{document} uniformly with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in [0, 1]$$\end{document} . By applying the estimates (5.64), (5.69), (5.77), (5.82), (5.85), (5.94) (using also that, by the ansatz (4.2), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert w_i \Vert _{s_0 + \Sigma ({\mathtt b})}\leqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 1,2$$\end{document} ) plus a repeated use of the triangular inequality and Lemma 2.6-(ii), one can show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{12} \textbf{B}$$\end{document} satisfies the estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Delta _{12} \textbf{B}|_{- M, s_0} \lesssim {\mathtt N}_n^{2 \tau + 1 + M} {\mathtt N}_{n - 1}^{- 2 {\mathtt a}} \varepsilon ^{M} \gamma ^{-M} \varepsilon _{M} \gamma ^{- (M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \\&\quad |\Delta _{12} \textbf{B}|_{- M, s_0 + {\mathtt b}} \lesssim {\mathtt N}_n^{2 \tau + 1 + M} {\mathtt N}_{n - 1}^{1 - {\mathtt a}} \varepsilon ^{M} \gamma ^{-M} \varepsilon _{M} \gamma ^{- (M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} . \end{aligned} \end{aligned}$$\end{document}

Collecting the estimates (5.95), (5.97), we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&| \Delta _{12} \textbf{E}_{n + 1} |_{- M, s_0} \lesssim \Big ( {\mathtt N}_n^{- {\mathtt b}} {\mathtt N}_{n - 1} + {\mathtt N}_n^{2 \tau + 1 + M} {\mathtt N}_{n - 1}^{- 2 {\mathtt a}} \varepsilon ^{M} \gamma ^{- M}\Big ) \varepsilon _{M} \gamma ^{-(M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \\&\quad | \Delta _{12} \textbf{E}_{n + 1} |_{- M, s_0 + {\mathtt b}} \lesssim \Big ( {\mathtt N}_{n - 1} + {\mathtt N}_n^{2 \tau + 1 + M} {\mathtt N}_{n - 1}^{1 - {\mathtt a}} \varepsilon ^{M} \gamma ^{-M} \ \Big ) \varepsilon _{M} \gamma ^{-(M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}, \end{aligned} \end{aligned}$$\end{document}

which imply the claimed bounds (5.69) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} , arguing as in (5.90)-(5.93).

We now verify the estimates (5.70) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . Recalling (5.87) and by Lemma 2.6-(v), together with the estimate (5.69), we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Delta _{12}({\mathtt z}_{n + 1}(j) - {\mathtt z}_n(j))|&= |\Delta _{12} {\widehat{\textbf{E}}}_n(0)_j^j| \lesssim |j|^{- M} |\Delta _{12} \textbf{E}_n|_{- M, s_0} \\&\lesssim |j|^{- M} {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon _{M} \gamma ^{-(M-1)} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} , \end{aligned} \end{aligned}$$\end{document}

which is the first estimate in (5.70) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} . Moreover, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt z}_{n + 1}(j) = {\mathtt z}_0(j) + \sum _{k = 0}^n {\mathtt z}_{k + 1} - {\mathtt z}_k$$\end{document} and recalling also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Delta _{12} {\mathtt z}_0| \lesssim \lambda ^{\theta } |j|^{- 1} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}$$\end{document} , we have,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\Delta _{12} {\mathtt z}_{n + 1}(j)|&\leqslant | \Delta _{12}{\mathtt z}_0(j)| + \sum _{k = 0}^n | \Delta _{12}({\mathtt z}_{k + 1} - {\mathtt z}_k) | \\&\lesssim \Big ( \lambda ^{\theta } |j|^{- 1} + |j|^{- M} \sum _{k = 0}^n {\mathtt N}_{k - 1}^{- {\mathtt a}} \varepsilon _{M} \gamma ^{-(M-1)} \Big ) \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} \\&\lesssim \lambda ^{\theta } |j|^{- 1} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} , \end{aligned} \end{aligned}$$\end{document}

by using (5.58), the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > 1$$\end{document} , and that the series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k \geqslant 0} {\mathtt N}_{k - 1}^{- {\mathtt a}} < \infty $$\end{document} is convergent and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \gamma ^{-1}\ll 1$$\end{document} . Hence, the second estimate in (5.70) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} is proved and the proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S3)}_{n + 1}$$\end{document} is completed.

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)}_{n + 1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega \in \Omega _{n+1}^{\gamma }(w_1) $$\end{document} . By (5.61) and the induction hypothesis (5.71) (i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)}_n$$\end{document} ), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Omega _{n+1}^{\gamma }(w_1) \subseteq \Omega _{n}^{\gamma }(w_1) \subseteq \Omega _{n}^{\gamma - \rho }(w_2) $$\end{document} . Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega \in \Omega _n^{\gamma - \rho }(w_2) \subseteq \Omega _n^{\gamma /2}(w_2) $$\end{document} because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \leqslant \gamma /2$$\end{document} . Thus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{n + 1}^\gamma (w_1) \subseteq \Omega _n^{\gamma - \rho }(w_2) \subseteq \Omega _n^{\gamma /2}(w_2) $$\end{document} . We also have that (5.60) holds for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \mathtt \Omega _{n+1}^{\gamma }(w_1)$$\end{document} , as well estimate (5.70) for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} , namely, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega \in \Omega _{n + 1}^\gamma (w_1)$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\Delta _{12} \mu _n(j; \omega )| = |\Delta _{12} {\mathtt z}_n(j; \omega )| \lesssim \lambda ^{\theta } |j|^{- 1} \Vert w_1 - w_2 \Vert _{ s_0 + \Sigma ({\mathtt b})}^\textrm{sup}. \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^{\nu }\setminus \{0\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j,j'\in {\mathbb {Z}}^2\setminus \{0\}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell |\leqslant {\mathtt N}_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell )+j-j' =0$$\end{document} . We distinguish two regimes:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{min}\{ |j|, |j'| \} \geqslant {\mathtt N}_n^\tau \quad \text {and} \quad \textrm{min}\{ |j|, |j'| \} \leqslant {\mathtt N}_n^\tau . \end{aligned}$$\end{document}

We then organize the rest of the proof in two claims.

Claim 1. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n+1}^{\gamma }(w_1)$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{|j|, |j'|\} \geqslant {\mathtt N}_{n}^\tau $$\end{document} , having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \ll 1 $$\end{document} small enough, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\textrm{i}\,\lambda \, \omega \cdot \ell + \mu _n(j;w_2) - \mu _n(j'; w_2)| \geqslant \frac{\lambda (\gamma - \rho )}{\langle \ell \rangle ^\tau |j'|^\tau }. \end{aligned}$$\end{document}

Proof of the Claim 1. By (5.60) and the estimates (5.62), we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in {\mathbb {Z}}^2 {\setminus }\{0\}$$\end{document} , that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\mu _n(j; w_2)| \leqslant C \lambda ^{\theta } |j|^{- 1} $$\end{document} for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C > 0$$\end{document} . Therefore, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \geqslant {\mathtt N}_n^\tau $$\end{document} and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D} {\mathtt C}(2 \gamma , \tau )$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _n(j; w_2) - \mu _n(j';w_2)|&\geqslant \lambda |\omega \cdot \ell | - C \lambda ^{\theta } \big ( |j|^{- 1} + |j'|^{- 1} \big ) \\&\geqslant \frac{2 \lambda \, \gamma }{\langle \ell \rangle ^\tau } - \frac{2 C \lambda ^{\theta }}{\textrm{min}\{ |j|, |j'| \}} \\&\geqslant \frac{2\lambda \,\gamma }{\langle \ell \rangle ^\tau } - \frac{2 C \lambda ^{\theta }}{{\mathtt N}_n^\tau } \geqslant \frac{\lambda \,\gamma }{\langle \ell \rangle ^\tau } , \end{aligned} \end{aligned}$$\end{document}

provided that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2C \lambda ^{\theta -1} \gamma ^{- 1} \langle \ell \rangle ^\tau {\mathtt N}_n^{- \tau } \leqslant 1 , \quad \forall \, \ell \in {\mathbb {Z}}^\nu , \quad 0 < |\ell | \leqslant {\mathtt N}_n. \end{aligned}$$\end{document}

The latter condition is verified by taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1} \gamma ^{- 1} \ll 1$$\end{document} small enough, by (4.7). The claimed statement then follows directly by (5.100) since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \geqslant \gamma - \rho $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \ell \rangle ^\tau \leqslant \langle \ell \rangle ^\tau |j'|^\tau $$\end{document} .

We now analyze the other regime in (5.99), namely when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min} \{ |j|, |j'| \} \leqslant {\mathtt N}_n^\tau $$\end{document} .

Claim 2. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n+1}^{\gamma }(w_1)$$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{|j|, |j'|\} \leqslant {\mathtt N}_{n}^\tau $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathtt N}_n^{\tau + \tau ^2} \lambda ^{\theta -1} \rho ^{- 1} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}\ll 1 $$\end{document} is small enough, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\textrm{i}\,\lambda \, \omega \cdot \ell + \mu _n(j; w_2) - \mu _n(j'; w_2)| \geqslant \frac{\gamma - \rho }{\langle \ell \rangle ^\tau |j'|^\tau }. \end{aligned}$$\end{document}

Proof of the Claim 2. By (5.98), and using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n+1}^{\gamma }(w_1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \min \{|j|, |j'|\} \leqslant {\mathtt N}_{n}^\tau $$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _n(j; w_2) - \mu _n(j'; w_2)| \geqslant |\textrm{i}\,\lambda \, \omega \cdot \ell + \mu _n(j; w_1) - \mu _n(j'; w_1)| \\&\qquad - |\Delta _{12} \mu _n(j)| - |\Delta _{12} \mu _{n}(j')| \\&\quad \geqslant \frac{\lambda \gamma }{\langle \ell \rangle ^\tau |j'|^{\tau }} - C \lambda ^{\theta } \big (|j|^{- 1} + |j'|^{- 1}\big ) \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} \\&\quad \geqslant \frac{\lambda \gamma }{\langle \ell \rangle ^\tau |j'|^{\tau }} - 2 C \lambda ^{\theta } \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})}\geqslant \frac{\lambda (\gamma - \rho )}{\langle \ell \rangle ^\tau |j'|^{\tau }} , \end{aligned} \end{aligned}$$\end{document}

provided the following condition holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2 C \lambda ^{\theta -1} \rho ^{- 1} \langle \ell \rangle ^\tau |j'|^\tau \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} \leqslant 1. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \ell \rangle \leqslant {\mathtt N}_n$$\end{document} and the momentum conservation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|j - j'| \lesssim |\ell | \lesssim {\mathtt N}_n$$\end{document} , one has that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |j'| \leqslant \textrm{min} \{ |j|, |j'| \} + |j - j'| \lesssim {\mathtt N}_n^\tau + {\mathtt N}_n \lesssim {\mathtt N}_n^\tau . \end{aligned}$$\end{document}

Hence the condition (5.102) is implied by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K \lambda ^{\theta -1} \rho ^{- 1} {\mathtt N}_n^{\tau + \tau ^2} \Vert w_1 - w_2 \Vert _{s_0 + \Sigma ({\mathtt b})} \leqslant 1 \end{aligned}$$\end{document}

for some large constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \gg 1$$\end{document} , and the claimed estimate (5.101) holds.

Finally, Claims 1 and 2 imply that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^\gamma (w_1)$$\end{document} , then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu {\setminus }\{0\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\ell | \leqslant {\mathtt N}_n$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | \lambda \, \omega \cdot \ell + \mu _n(j; w_2) - \mu _n(j'; w_2) | \geqslant \frac{\lambda (\gamma - \rho )}{\langle \ell \rangle ^\tau |j'|^\tau } , \end{aligned}$$\end{document}

which is exactly the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^{\gamma - \rho }(w_2)$$\end{document} . The proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)}_{n+1}$$\end{document} is then concluded. This concludes also the proof of Proposition 5.9. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

KAM reducibility: convergence

The eigenvalues of the normal form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_{n}$$\end{document} in (5.60) are Cauchy sequences and so they converge to some limit.

Lemma 5.11

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 \setminus \{ 0 \}$$\end{document} , the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \widetilde{\mu }_n(j) \}_{n\in {\mathbb {N}}}$$\end{document} , defined in (5.60), converges to some limit

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _\infty (j) = \textrm{i}\, \beta \,{\mathtt L}(j) + {\mathtt z}_\infty (j), \quad \mu _\infty (j)= \mu _\infty (j;\,\cdot \,):{{\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}}\rightarrow {\mathbb {C}},\nonumber \\ \end{aligned}$$\end{document}

satisfying the following estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&| \mu _\infty (j) - {{\widetilde{\mu }}}_n(j) |^{\textrm{Lip}(\gamma )} = | {\mathtt z}_\infty (j) - {{\widetilde{{\mathtt z}}}}_n(j) |^{\textrm{Lip}(\gamma )} \lesssim {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon _{M} \gamma ^{-(M-1)} |j|^{- M} , \\&\quad | {\mathtt z}_\infty (j)|^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{\theta } |j|^{- 1} . \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

In addition, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\infty (j;\,\cdot \,):{{\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}}\rightarrow \textrm{i}\,{\mathbb {R}}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} .

Proof

By Proposition 5.9 and by (5.68), we have that the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\widetilde{\mu }_n(j;\omega )\}_{n\in {\mathbb {N}}}\subset {\mathbb {C}}$$\end{document} is Cauchy on the closed set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} , therefore it is convergent for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathtt D}{\mathtt C}(2\gamma ,\tau )\cap \Lambda _{o}$$\end{document} . The estimates in (5.104) follow then by a telescoping argument with (5.68). Finally, if we assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\widetilde{\mu }_n(j;\omega )\}_{n\in {\mathbb {N}}}\subset \textrm{i}\,{\mathbb {R}}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} , by (5.67), implying that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu _\infty (j;\omega ) \in \textrm{i}\,{\mathbb {R}}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We define the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^\gamma $$\end{document} of the non-resonance conditions for the final eigenvalues as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Omega _\infty ^\gamma := \Omega _\infty ^\gamma (w):= \Big \{ \omega \in {{\mathtt D}{\mathtt C}(2 \gamma ,\tau ) \cap \Lambda _{o}} \,: \, |\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _\infty (j) - \mu _\infty (j')| \geqslant \frac{2 \lambda \,\gamma }{\langle \ell \rangle ^\tau | j' |^\tau } , \\&\quad \ \ \forall \ell \in {\mathbb {Z}}^\nu \setminus \{ 0 \}, \ \ j,j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}, \ \ \pi ^\top (\ell ) + j-j' =0\Big \}. \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

Lemma 5.12

We have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^\gamma \subseteq \cap _{n \geqslant 0} \, \Omega _n^\gamma $$\end{document} .

Proof

We prove by induction that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^\gamma \subseteq \Omega _n^\gamma $$\end{document} for any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} . The statement is trivial for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _0^\gamma := {\mathtt D}{\mathtt C}(2 \gamma ,\tau )\cap \Lambda _{o}$$\end{document} (see Proposition 5.9).

We now assume by induction that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^\gamma \subseteq \Omega _n^\gamma $$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} and we show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^\gamma \subseteq \Omega _{n + 1}^\gamma $$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {Z}}^\nu {\setminus } \{ 0 \}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j, j' \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top ( \ell ) +j-j'=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_n$$\end{document} . By (5.104), (5.105), we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} |\textrm{i}\, \lambda \,\omega \cdot \ell + \mu _n(j) - \mu _n(j')|&\geqslant |\textrm{i}\, \lambda \,\omega \cdot \ell + \mu _\infty (j) - \mu _\infty (j')| - |\mu _\infty (j) - \mu _n(j)| \\&\quad - |\mu _\infty (j') - \mu _n(j')| \\&\geqslant \frac{2\lambda \,\gamma }{\langle \ell \rangle ^\tau | j' |^\tau } - C {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon _{M} \gamma ^{-(M-1)}\big ( |j|^{- M} + |j'|^{- M} \big ) \\&\geqslant \frac{\lambda \,\gamma }{\langle \ell \rangle ^\tau | j' |^\tau } , \end{aligned} \end{aligned}$$\end{document}

for some positive constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , provided

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C {\mathtt N}_{n - 1}^{- {\mathtt a}}\varepsilon ^{M}\gamma ^{- M}\langle \ell \rangle ^\tau \big ( | j' |^\tau |j|^{- M} + |j'|^{\tau - M} \big ) \leqslant 1. \end{aligned}$$\end{document}

Recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \tau $$\end{document} by (5.56), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|j'|^{\tau - M} \leqslant 1$$\end{document} . Moreover, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\ell | \leqslant {\mathtt N}_n$$\end{document} and the momentum condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j-j'=0$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|j'| \leqslant |j| + |j - j'| \leqslant |j| + {\mathtt N}_n \lesssim {\mathtt N}_n |j|, \end{aligned} \end{aligned}$$\end{document}

and therefore, using again that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M > \tau $$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | j' |^\tau |j|^{- M} \lesssim {\mathtt N}_n^\tau |j|^{\tau - M} \lesssim {\mathtt N}_n^\tau . \end{aligned}$$\end{document}

We then deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C {\mathtt N}_{n - 1}^{- {\mathtt a}}\varepsilon ^{M}\gamma ^{- M}\langle \ell \rangle ^\tau \Big ( | j' |^\tau |j|^{- M} + |j'|^{\tau - M} \Big ) \leqslant C_0 {\mathtt N}_n^{2 \tau } {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^{M} \gamma ^{- M} \end{aligned}$$\end{document}

for some large constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0 \gg 0$$\end{document} . Hence, the condition (5.106) is verified since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_0 {\mathtt N}_n^{2 \tau } {\mathtt N}_{n - 1}^{- {\mathtt a}} \varepsilon ^{M} \gamma ^{- M} \leqslant 1, \end{aligned}$$\end{document}

recalling (5.56), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \gamma ^{-1} = \lambda ^{\theta -1}\gamma ^{-1}\ll 1$$\end{document} in (4.7) and the smallness condition (5.59) We conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _{n + 1}^\gamma $$\end{document} and the claim is proved. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Now we define the sequence of invertible maps

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\widetilde{\Phi }}}_n:= \Phi _0 \circ \Phi _1 \circ \ldots \circ \Phi _n , \quad n \in {\mathbb {N}}. \end{aligned}$$\end{document}

Proposition 5.13

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + \Sigma ({\mathtt b})$$\end{document} . There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau ,\nu ) > 0$$\end{document} such that, if (4.2) (5.59) are verified, then the following holds. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma $$\end{document} , the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\widetilde{\Phi }}}_n)_{n\in {\mathbb {N}}}$$\end{document} converges in norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$| \cdot |_{0, s}^{\textrm{Lip}(\gamma )}$$\end{document} to an invertible map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _\infty $$\end{document} , satisfying, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0\leqslant s\leqslant S-\Sigma ({\mathtt b})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\Phi _\infty ^{\pm 1} - {{\widetilde{\Phi }}}_n^{\pm 1}|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s} {\mathtt N}_{n + 1}^{2 \tau + 1} {\mathtt N}_n^{- {\mathtt a}} \varepsilon ^{M} \gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} , \\&\quad |\Phi _\infty ^{\pm 1} - \textrm{Id}|_{0, s}^{\textrm{Lip}(\gamma )} \lesssim _{s} {\mathtt N}_{0}^{2\tau +1}\varepsilon ^{M} \gamma ^{- M} \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} . \end{aligned} \end{aligned}$$\end{document}

The operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _\infty ^{\pm 1}: H^s_0 \rightarrow H^s_0$$\end{document} are real and momentum preserving. Moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma $$\end{document} , one has the conjugation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textbf{L}}_{\infty }:= \Phi _\infty ^{- 1} {\textbf{L}}_{0} \Phi _\infty =\lambda \, \omega \cdot \partial _\varphi + {\textbf{D}}_\infty , \quad {\textbf{D}}_\infty := \textrm{diag}_{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} \mu _\infty (j) , \end{aligned}$$\end{document}

where the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_{0}$$\end{document} is given in (5.55)

and the final eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\infty (j)$$\end{document} are given in Lemma 5.11. In addition, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w({\varphi },x)=\textrm{odd}({\varphi },x)$$\end{document} , the operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{\infty }^{\pm 1}$$\end{document} are reversibility preserving and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_\infty $$\end{document} is a reversible operator.

Proof

The existence of the invertible map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _\infty ^{\pm 1}$$\end{document} and the estimates (5.108) follow by (5.66), (5.107), arguing as in Corollary 4.1 in [6]. By (5.107), Lemma 5.12 and Proposition 5.9, one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\widetilde{\Phi }}}_n^{- 1} {\textbf{L}}_0 {{\widetilde{\Phi }}}_n =\lambda \, \omega \cdot \partial _\varphi + {\textbf{D}}_n + {\textbf{E}}_n$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} . The claimed statement then follows by passing to the limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} , by using (5.64), (5.108) and Lemma 5.11. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Inversion of the Linearized Operator \documentclass[12pt]{minimal}

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			\begin{document}$${{\mathcal {L}}}$$\end{document}L

In this section we show that the linearized operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}$$\end{document} in (4.3) is invertible and we prove tame estimates for its inverse. By Propositions 5.1, 5.5, 5.6, 5.13, using also Lemma 2.6-(i), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma $$\end{document} , we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {W}}}_\infty := {\mathcal {B}}_{\perp } \circ \mathbf{\Phi }_M \circ \Phi _\infty \end{aligned}$$\end{document}

and we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {W}}}_\infty ^{- 1} {{\mathcal {L}}} {{\mathcal {W}}}_\infty = {\textbf{L}}_{\infty } , \quad \forall \,\omega \in \Omega _\infty ^\gamma , \end{aligned}$$\end{document}

with the maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {W}}}_\infty \,,\, {{\mathcal {W}}}_\infty ^{- 1}: H^s_0({\mathbb {T}}^{\nu + 2}) \rightarrow H^s_0({\mathbb {T}}^{\nu + 2})$$\end{document} are real and momentum preserving and, if (4.2) holds with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0 = \Sigma (\mathtt b)$$\end{document} , they satisfy for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S-\Sigma ({\mathtt b})$$\end{document} , the bounds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {W}}}_\infty ^{\pm 1} h \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _s \Vert h \Vert _s^{\textrm{Lip}(\gamma )} + \Vert w \Vert _{s + \Sigma ({\mathtt b})}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0}^{\textrm{Lip}(\gamma )}. \end{aligned}$$\end{document}

In addition, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {W}}}_\infty , {{\mathcal {W}}}_\infty ^{- 1}$$\end{document} are reversibility preserving.

Now we define the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\infty ^\gamma $$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Lambda _\infty ^\gamma := \Lambda _\infty ^\gamma (w):= \Big \{ \omega \in {{\mathtt D}{\mathtt C}(2 \gamma , \tau ) \cap \Lambda _{o}} \,: \, |\textrm{i}\,\lambda \,\omega \cdot \ell + \mu _\infty (j) | \geqslant \frac{2 \lambda \, \gamma }{\langle \ell \rangle ^\tau } , \\&\quad \ \forall \, \ell \in {\mathbb {Z}}^\nu \setminus \{0\} ,\ \ j \in {\mathbb {Z}}^2 \setminus \{ 0 \}, \ \ \pi ^\top (\ell ) + j = 0 \Big \}. \end{aligned} \end{aligned}$$\end{document}

First, we discuss the invertibility of the diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\infty $$\end{document} given in (5.109).

Lemma 6.1

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Lambda _\infty ^\gamma $$\end{document} , the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\infty (\omega )$$\end{document} is invertible and its inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\infty ^{- 1}$$\end{document} satisfies the bound, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \textbf{L}_\infty ^{- 1} h \Vert _s^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{- 1} \gamma ^{- 1} \Vert h \Vert _{s + 2 \tau + 1}^{\textrm{Lip}(\gamma )} . \end{aligned}$$\end{document}

In addition, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\infty ^{- 1}$$\end{document} is reversible.

Proof

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Lambda _\infty ^\gamma $$\end{document} , the inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_\infty (\omega )^{- 1}$$\end{document} is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{L}_\infty (\omega )^{- 1} h(\varphi , x) = \sum _{ \ell \in {\mathbb {Z}}^\nu \setminus \{0\},\, j \in {\mathbb {Z}}^2 \setminus \{ 0 \} \atop \pi ^\top (\ell ) + j = 0 } \frac{1}{\eta _{\ell j}(\omega )} \widehat{h}(\ell , j) e^{\textrm{i}\ell \cdot \varphi } e^{\textrm{i}j \cdot x} , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\ell j }(\omega ) = \textrm{i}\,\lambda \,\omega \cdot \ell + \mu _\infty (j;\omega )$$\end{document} and we used the momentum condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j = 0$$\end{document} to get the restriction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ne 0$$\end{document} , since we already have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \ne 0$$\end{document} . This clearly implies the bound, by (6.3), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \textbf{L}_\infty (\omega )^{- 1} h \Vert _s \lesssim \lambda ^{- 1} \gamma ^{- 1} \Vert h \Vert _{s + \tau }. \end{aligned}$$\end{document}

Moreover, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1, \omega _2 \in \Lambda _\infty ^\gamma $$\end{document} , by (6.3) Lemma 5.11 and using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1}\gamma ^{-1}\leqslant 1$$\end{document} in (4.7), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Big | \frac{1}{\eta _{\ell j}(\omega _1)} - \frac{1}{\eta _{\ell j}(\omega _2)} \Big |&\lesssim \frac{\lambda \,|\ell | |\omega _1 - \omega _2| + |\mu _\infty (j; \omega _1) - \mu _\infty (j; \omega _2)|}{|\eta _{\ell j}(\omega _1)| |\eta _{\ell j}(\omega _2)|} \\&\lesssim \langle \ell \rangle ^{2 \tau }\lambda ^{- 2} \gamma ^{- 2} \big ( \lambda \,|\ell | + \lambda ^{\theta } \gamma ^{- 1} \big ) |\omega _1 - \omega _2| \\&\lesssim \lambda ^{- 1} \gamma ^{- 2} \langle \ell \rangle ^{2 \tau + 1} |\omega _1 - \omega _2| , \end{aligned} \end{aligned}$$\end{document}

which implies that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big \Vert \big ( \textbf{L}_\infty (\omega _1)^{- 1} - \textbf{L}_\infty (\omega _2)^{- 1}\big ) h \big \Vert _s \lesssim \lambda ^{- 1} \gamma ^{- 2} \Vert h \Vert _{s + 2 \tau + 1} . \end{aligned}$$\end{document}

Now, using that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\omega ) \in H^{s + 2 \tau + 1}_0$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1, \omega _2 \in \Lambda _\infty ^\gamma $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \textbf{L}_\infty (\omega _1)^{- 1} h(\omega _1) - \textbf{L}_\infty (\omega _2)^{- 1} h(\omega _2) \Vert _s \\&\quad \leqslant \Vert \textbf{L}_\infty (\omega _1)^{- 1}\big ( h(\omega _1) - h(\omega _2) \big ) \Vert _s + \Vert \big ( \textbf{L}_\infty (\omega _1)^{- 1} - \textbf{L}_\infty (\omega _2)^{- 1} \big ) h(\omega _2) \Vert _s , \end{aligned} \end{aligned}$$\end{document}

then, together with the bounds (6.6), (6.7), we obtain the claimed estimate (6.4). Finally, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_{\infty }, \textbf{L}_\infty ^{- 1}$$\end{document} are reversible operators, by (6.5), Lemma 5.11 and Lemma 2.9. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We are now in position to state the following proposition on the invertibility of the linearized operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}$$\end{document} in (4.3).

Proposition 6.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\sigma }}:= \Sigma ({\mathtt b}) + 2 \tau + 1$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ({\mathtt b})$$\end{document} given in (5.56). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S > s_0 + {\overline{\sigma }}$$\end{document} . There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta := \delta (S, \tau , \nu ) > 0$$\end{document} such that, if (4.2) (5.59) are verified (with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = {\overline{\sigma }}$$\end{document} ), then the following holds. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma \cap \Lambda _\infty ^\gamma $$\end{document} the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}={{\mathcal {L}}}(\omega )$$\end{document} , defined (4.3), is invertible on the subspace of the quasi-periodic traveling waves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _\varsigma h({\varphi })=h({\varphi }-\pi (\varsigma ))$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varsigma \in {\mathbb {R}}$$\end{document} , and its inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^{-1}={{\mathcal {L}}}(\omega )^{- 1}$$\end{document} satisfies the tame bound, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant S-{\overline{\sigma }}$$\end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in H^{s + {\overline{\sigma }}}_0({\mathbb {T}}^{\nu +2})$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {L}}}^{- 1} h \Vert _s^{\textrm{Lip}(\gamma )} \lesssim _s \lambda ^{- 1} \gamma ^{- 1}\Big ( \Vert h \Vert _{s + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} + \Vert w \Vert _{s + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \Vert h \Vert _{s_0 + \overline{\sigma }}^{\textrm{Lip}(\gamma )} \Big ). \end{aligned}$$\end{document}

Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^{-1}h$$\end{document} is a quasi-periodic traveling wave. In addition, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\mathcal {L}}}^{- 1}$$\end{document} is reversible.

Proof

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \Omega _\infty ^\gamma \cap \Lambda _\infty ^\gamma $$\end{document} , one has that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}^{- 1} = {\mathcal W}_\infty ^{- 1} \textbf{L}_\infty ^{- 1} {{\mathcal {W}}}_\infty $$\end{document} . Hence, the estimate (6.8) follows by (6.2) and Lemma 6.1, using also the ansatz (4.2) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = {\overline{\sigma }}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{L}_{\infty }^{\pm 1}$$\end{document} are momentum preserving by (5.109), (6.5), and the maps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}_{\infty }^{\pm 1}$$\end{document} are momentum preserving by (6.1) and Propositions 5.1, 5.6, 5.13, then, by Lemmata 2.12, 2.13, we conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^{-1}h$$\end{document} is a quasi-periodic traveling wave, whenever is h. Finally, if we assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w = \textrm{odd}({\varphi },x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^{-1}$$\end{document} is reversible by Lemma 6.1 and by the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}_{\infty }^{\pm 1}$$\end{document} are reversibility preserving. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The Nash Moser scheme

In this section we construct the solution of the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal F}(v) = 0$$\end{document} in (3.1) by means of a Nash Moser nonlinear iteration. We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _n$$\end{document} the orthogonal projector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{{\mathtt N}_n}$$\end{document} (on quasi-periodic traveling waves, see (2.6)) on the finite dimensional space

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {H}}}_n:= \big \{ w \in L_0^2({\mathbb {T}}^{\nu + 2}, {\mathbb {R}}) \, \ \text { such that } (2.1.1) \text { holds } \, \,: \, \ w = { \Pi }_n w \big \} , \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _n^\bot := \textrm{Id} - \Pi _n$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_0^2({\mathbb {T}}^{\nu +2})=H_0^0({\mathbb {T}}^{\nu +2})$$\end{document} , see (1.10). The projectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Pi _n $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Pi _n^\bot $$\end{document} satisfy the usual smoothing properties in Lemma 2.4, namely

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \Pi _{n} v \Vert _{s+b}^{\textrm{Lip}(\gamma )} \leqslant {\mathtt N}_{n}^{b} \Vert v \Vert _{s}^{\textrm{Lip}(\gamma )} , \quad \ \Vert \Pi _{n}^\bot v \Vert _{s}^{\textrm{Lip}(\gamma )} \leqslant {\mathtt N}_n^{- b} \Vert v \Vert _{s + b}^{\textrm{Lip}(\gamma )} , \quad \ s,b \geqslant 0 . \end{aligned}$$\end{document}

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau , {\mathtt N}_{0}>0$$\end{document} , we define the constants

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{\mathtt N}_{n}:= {\mathtt N}_{0}^{\chi ^n}, \quad n \geqslant 0, \quad {\mathtt N}_{- 1}:= 1, \quad \chi = 3/2, \quad \kappa := 12( {\overline{\sigma }} + 1) + 1, \\&\quad \mathtt a_1:= \textrm{max}\{ 6 {\overline{\sigma }} + 13,\, \chi ^2(\tau + \tau ^2 + 1) + \chi (2 {\overline{\sigma }} + 1) + 1 \} , \\&\quad {\overline{\tau }}:= 2 {\overline{\sigma }} + 4 + \mathtt a_1 + \chi (\tau + \tau ^2 + 1)+\tau _{1}, \quad {\mathtt b}_1:= 2 {\overline{\sigma }} + 4 +\chi ^{-1} ( \mathtt a_1 + \kappa ). \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\sigma }} > 1$$\end{document} is given in Proposition 6.2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{1}>1$$\end{document} is fixed in (5.56).

Remark 7.1

Let us describe of the parameters in (7.2) and their role in the following Proposition 7.3. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt a}_{1}>0$$\end{document} appears in the negative exponent of the estimate in (7.8) and measures how the low regularity norm of the nonlinear functional is fastly decreasing to 0 at each approximate solution of the iteration. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt b}_{1}>0$$\end{document} appears in the regularity index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0+{\mathtt b}_{1}$$\end{document} of the second estimate in (7.9), which is the high regularity norms of the approximate solution and the functional evaluated at it that are allowed to diverge, with exponent rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa >0$$\end{document} . The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\tau }}>0$$\end{document} appears as an exponent in the smallness condition (7.4). The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\sigma }}>0$$\end{document} accounts for the loss of derivatives of the KAM iteration. The role of the parameters in (7.2) is comparable with the ones in (5.56) for the KAM reducibility of the linearized operators (see also Remark 5.8). The differences are in their definitions, due to the nonlinear iteration and the loss of derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\sigma }}>\Sigma ({\mathtt b})>0$$\end{document} (see Proposition 6.2), and the exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa >0$$\end{document} in the divergence of the high regularity norms.

Note that Proposition 3.3 provides an approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app},N}$$\end{document} which is a quasi periodic traveling wave such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\geqslant 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v_{\textrm{app},N}\Vert _{s}^{\textrm{Lip}(\gamma )}$$\end{document} is of size 1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\mathcal F}(v_{\textrm{app},N}) \Vert _s^{\textrm{Lip}(\gamma )}$$\end{document} is uniformly bounded with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} . We state these properties as follows.

Lemma 7.2

(Initialization of the Nash-Moser iteration). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in {\mathbb {N}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,1)$$\end{document} as in (3.15). For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant 0$$\end{document} , there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(s)>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert {{\mathcal {F}}}(v_{\textrm{app},N}) \Vert _s^{\textrm{Lip}(\gamma )} \leqslant C(s),\quad \Vert v_{\textrm{app},N} \Vert _s^{\textrm{Lip}(\gamma )} \leqslant C(s), \\&\quad \inf _{\omega \in \Omega _\gamma } \Vert v_{\textrm{app, N}}(\cdot ; \omega )\Vert _s \geqslant K(s) \lambda ^{- \mathtt c} \end{aligned} \end{aligned}$$\end{document}

uniformly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \geqslant {\overline{\lambda }} \gg 1$$\end{document} sufficiently large, for some constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(s), K(s) > 0$$\end{document} . In addition, if the forcing term f in (3.1) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\textrm{app, N}} = \textrm{odd}({\varphi }, x)$$\end{document} .

The lower bound in (7.3), that has been proved in Proposition 3.3, is actually not needed in the nonlinear iteration. It will be used to prove Theorem 1.3 in the next section.

We now prove the following proposition.

Proposition 7.3

(Nash-Moser). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau > 0$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\gamma }$$\end{document} be defined in (3.6) With the notation in (7.2), there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta \in (0, 1)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_* > 0$$\end{document} such that if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{\mathtt N}_{0}^{{\overline{\tau }}} \lambda ^{- 1} \leqslant \delta , \quad {\mathtt N}_{0}:= \gamma ^{- 1}, \quad \gamma := \lambda ^{- \mathtt c}, \quad \text {for some} \ \ \ 0< \mathtt c < \textrm{min} \{ {\overline{\tau }}\,^{-1}, \tfrac{1}{3}(2-\alpha ) \}, \end{aligned}\nonumber \\ \end{aligned}$$\end{document}

then the following properties hold for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1)_{n}$$\end{document} There exist a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0 > 0$$\end{document} large enough and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n - w_0: {{\mathcal {G}}}_n \rightarrow {{\mathcal {H}}}_{n - 1} $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_0:= v_{\textrm{app},N}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {H}}}_{- 1}:= \{ 0 \}$$\end{document} , satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_{n} \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \leqslant C_0 . \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , the difference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ h_n:= w_n - w_{n - 1}$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert h_1 \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \lesssim \lambda ^{- 1} \gamma ^{-1}$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert h_n \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \leqslant C_* {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} \lambda ^{- 1} . \end{aligned}$$\end{document}

The sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ {{\mathcal {G}}}_n\}_{n \geqslant 0}$$\end{document} are defined as follows. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_{0}$$\end{document} we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_0:= \Omega _\gamma $$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {G}}}_{n + 1}:= {{\mathcal {G}}}_n \cap \big ( \Omega _\infty ^{\gamma _n}(w_n) \cap \Lambda _\infty ^{\gamma _n}(w_n) \big ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma _{n}:=\gamma (1 + 2^{-n}) $$\end{document} and the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^{\gamma _n}(w_n)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\infty ^{\gamma _n}(w_n)$$\end{document} are defined in (5.105), (6.3), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{o}:=\mathcal {G}_{n}$$\end{document} . Moreover, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n({\varphi },x;\omega )$$\end{document} is a quasi-periodic traveling wave. In addition, if the forcing term f in (3.1) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{odd}({\varphi }, x)$$\end{document} ;

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}2)_{n}$$\end{document} On the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n$$\end{document} , we have the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_{0}}^{\textrm{Lip}(\gamma )} \leqslant C_* {\mathtt N}_{n - 1 }^{- \mathtt a_1}\,; \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}3)_{n}$$\end{document} On the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n$$\end{document} , we have the estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_n \Vert _{s_0 + \mathtt b_1}^{\textrm{Lip}(\gamma )}+ \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + \mathtt b_1}^{\textrm{Lip}(\gamma )} \leqslant C_* {\mathtt N}_{n-1}^{\kappa } . \end{aligned}$$\end{document}

Proof

To simplify notations, in this proof we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _s$$\end{document} instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _s^{\textrm{Lip}(\gamma )}$$\end{document} .

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1, 2, 3)_0$$\end{document} . By (7.3) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=s_0,s_0+{\mathtt b}_{1}$$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert w_0\Vert _{s} = \Vert v_{\textrm{app},N} \Vert _{s} \leqslant C(s)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {{\mathcal {F}}} (w_0 ) \Vert _s = \Vert {{\mathcal {F}}} (v_{\textrm{app},N} ) \Vert _s \leqslant C(s)$$\end{document} . Then (7.5), (7.8) and (7.9) hold taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{1}{2} C_0, \tfrac{1}{2}C_*(s) \geqslant C(s)$$\end{document} sufficiently large. In particular, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_0 \Vert _{s_0+{\overline{\sigma }}} \leqslant \tfrac{1}{2} C_0 \leqslant C_0. \end{aligned}$$\end{document}

Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1,2,3)_n$$\end{document} hold for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} , and prove \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1,2,3)_{n+1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{o}:=\mathcal {G}_{n}$$\end{document} . By \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1)_n$$\end{document} , one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert w_n\Vert _{s_0 + {\overline{\sigma }}} \leqslant C_0$$\end{document} , for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0 \gg 0$$\end{document} large enough, independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} . The assumption (7.4) implies the smallness condition (5.59) of Proposition 5.9. Indeed, by the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\tau }}$$\end{document} in (7.2), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\overline{\tau }} > \tau _{1} +1 , \quad \text {that is} \quad \tau _{1}<{\overline{\tau }}-1 . \end{aligned}$$\end{document}

Moreover, by (7.4), (5.58), (4.5), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \gamma ^{-1}=(\lambda ^{-1})^{2(1-{\mathtt c})-\alpha }\ll 1$$\end{document} , which implies, using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M>\frac{1-{\mathtt c}}{2(1-{\mathtt c})-\alpha }$$\end{document} by (5.56),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon ^M \gamma ^{-M} = (\lambda ^{-1})^{(2(1-{\mathtt c})-\alpha )M} < (\lambda ^{-1})^{1-{\mathtt c}} . \end{aligned}$$\end{document}

We conclude that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_{0}>1$$\end{document} satisfies (7.4) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt N}_{0}^{\tau _{1}} \varepsilon ^M \gamma ^{-M}< {\mathtt N}_{0}^{{\overline{\tau }}} {\mathtt N}_{0}^{-1} \varepsilon ^M \gamma ^{-M} < {\mathtt N}_{0}^{{\overline{\tau }}} {\mathtt N}_{0}^{-1}(\lambda ^{-1})^{1-{\mathtt c}} = {\mathtt N}_{0}^{{\overline{\tau }}} \lambda ^{-{\mathtt c}} \lambda ^{{\mathtt c}-1} = {\mathtt N}_{0}^{{\overline{\tau }}} \lambda ^{-1} \leqslant \delta , \end{aligned}$$\end{document}

that is, (5.59) is satisfied as well, as claimed. Then Proposition 6.2 applies to the linearized operator

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {L}}}_n \equiv {{\mathcal {L}}}(w_n): = \textrm{d}_{v} {\mathcal F}(w_n) . \end{aligned}$$\end{document}

This implies that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_{n + 1}$$\end{document} , the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_n(\omega )$$\end{document} admits a right inverse \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal L}_n(\omega )^{- 1}$$\end{document} satisfying the tame estimates, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0 \leqslant s \leqslant s_0 + {\mathtt b}_{1}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {L}}}_n^{-1} h \Vert _s \lesssim _s \lambda ^{- 1}\gamma ^{-1} \big ( \Vert h \Vert _{s + {\overline{\sigma }}} + \Vert w_n \Vert _{s + \overline{\sigma }} \, \Vert h \Vert _{s_0 + {\overline{\sigma }}} \big ), \end{aligned}$$\end{document}

using the bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _n = \gamma (1 + 2^{- n}) \in [\gamma , 2\gamma ]$$\end{document} . By (7.11) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s= s_0$$\end{document} and (7.5), one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {{\mathcal {L}}}_n^{- 1} h \Vert _{s_0} \lesssim \lambda ^{- 1}\gamma ^{- 1} \Vert h \Vert _{s_0 + {\overline{\sigma }}}. \end{aligned}$$\end{document}

We define the successive approximation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_{n + 1}:= w_n + h_{n + 1} , \quad {h}_{n + 1}:= - \Pi _n {{\mathcal {L}}}_n^{-1} \Pi _n {{\mathcal {F}}}(w_n) \in {{\mathcal {H}}}_{n} , \end{aligned}$$\end{document}

defined for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_{n + 1}$$\end{document} , and the remainder

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_n:= {{\mathcal {F}}}(w_{n+1}) - {{\mathcal {F}}}(w_n) - {{\mathcal {L}}}_n h_{n+1}. \end{aligned}$$\end{document}

We now estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{n + 1}$$\end{document} . Note that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s, \mu \geqslant 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert w_n \Vert _{s+\mu } \leqslant \Vert w_0 \Vert _{s + \mu } + \Vert w_n - w_0 \Vert _{s + \mu } {\mathop {\lesssim _{s, \mu }}\limits ^{w_0 = v_{\textrm{app, N}},\, (7.2)}} 1 + {\mathtt N}_{n - 1}^\mu \Vert w_n - w_0 \Vert _s \\&\quad \lesssim _{s, \mu } {\mathtt N}_{n - 1}^\mu (1 + \Vert w_n \Vert _s)\, \quad \text {and hence by } (7), \\&\quad \Vert {{\mathcal {L}}}_n^{- 1} h \Vert _{s_0 + 1} \lesssim {\mathtt N}_{n - 1}\lambda ^{- 1}\gamma ^{- 1} \Vert h \Vert _{s_0 + {\overline{\sigma }} + 1} \end{aligned} \end{aligned}$$\end{document}

By the estimates (7.11), (7.12), (7.5), (7.1), (7.2), (7.4), (7.8), (7.14), by Lemma (3.1)-(i), we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert h_{n + 1} \Vert _{s_0}&\lesssim \lambda ^{- 1}\gamma ^{- 1} \Vert \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0 + {\overline{\sigma }}} \lesssim {\mathtt N}_n^{{\overline{\sigma }}} \lambda ^{- 1} \gamma ^{- 1} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0} , \\ \Vert h_{n + 1} \Vert _{s_0 + {\overline{\sigma }}}&\lesssim {\mathtt N}_n^{{\overline{\sigma }}} \Vert {{\mathcal {L}}}_n^{-1} \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0} \lesssim {\mathtt N}_n^{2 {\overline{\sigma }}} \lambda ^{- 1} \gamma ^{- 1} \Vert {\mathcal F}(w_n) \Vert _{s_0}. \end{aligned} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert h_{n + 1} \Vert _{s_0 + \mathtt b_1}&\lesssim \Vert {\mathcal {L}}_{n}^{-1} \Pi _{n} {\mathcal {F}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} \\&\lesssim \lambda ^{- 1} \gamma ^{- 1} \Big ( \Vert \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1 + {\overline{\sigma }}} + \Vert w_n \Vert _{s_0 + \mathtt b_1 + {\overline{\sigma }}} \Vert \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0 + {\overline{\sigma }}} \Big ) \\&\lesssim \lambda ^{- 1} \gamma ^{- 1} {\mathtt N}_n^{2 {\overline{\sigma }} } \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0} \Big ) \\&\lesssim \lambda ^{- 1} {\mathtt N}_n^{2 {\overline{\sigma }} +1} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} +C_* {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big ) \\&\lesssim \lambda ^{- 1} {\mathtt N}_n^{2 {\overline{\sigma }} +1} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big ). \end{aligned} \end{aligned}$$\end{document}

Moreover, (7.13) and estimate (7.16), using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{-1}\ll 1$$\end{document} , imply that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert w_{n + 1} \Vert _{s_0 + \mathtt b_1}&\leqslant \Vert w_n \Vert _{s_0 + \mathtt b_1} + \Vert h_{n + 1} \Vert _{s_0 + \mathtt b_1} \\&\lesssim {\mathtt N}_{n}^{2 {\overline{\sigma }} + 1} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big ). \end{aligned} \end{aligned}$$\end{document}

Next, we estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {{\mathcal {F}}}(w_{n + 1}) \Vert _{s_0}$$\end{document} . By the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{n+1}$$\end{document} in (7.13), recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{n} + \Pi _{n}^\perp =\textrm{Id}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _{n}^\perp \Pi _{n} = 0$$\end{document} , we obtain that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_{n + 1}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {{\mathcal {F}}}(w_{n + 1})&= {{\mathcal {F}}}(w_n) + {{\mathcal {L}}}_n h_{n + 1} + Q_n \\&= {\mathcal {F}}(w_n) - {\mathcal {L}}_{n} (\textrm{Id} - \Pi _{n}^\perp ) {\mathcal {L}}_{n}^{-1} \Pi _{n} {\mathcal {F}}(w_n) + Q_n \\&= \Pi _n^\bot {{\mathcal {F}}}(w_n) + {{\mathcal {L}}}_n \Pi _n^\bot {{\mathcal {L}}}_n^{- 1} \Pi _n {{\mathcal {F}}}(w_n) + Q_n \\&= \Pi _n^\bot {{\mathcal {F}}}(w_n) + [{{\mathcal {L}}}_n, \Pi _n^\bot ]{{\mathcal {L}}}_n^{- 1} \Pi _n {{\mathcal {F}}}(w_n) + Q_n . \end{aligned} \end{aligned}$$\end{document}

We estimate separately the three terms in (7.18). Note that, by (7.4), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{- 1} = {\mathtt N}_{0} \leqslant {\mathtt N}_{n}$$\end{document} . By (7.1), Lemma 3.1-(i), (7.5) and (7.4), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert \Pi _n^\bot {{\mathcal {F}}}(w_n) \Vert _{s_0} \leqslant {\mathtt N}_{n}^{- \mathtt b_1} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1 } , \quad \Vert \Pi _n^\bot {{\mathcal {F}}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} \leqslant \Vert {\mathcal F}(w_n) \Vert _{s_0 + \mathtt b_1 } . \end{aligned} \end{aligned}$$\end{document}

By (7.5), Lemma 3.1-(i),(ii), (7.1), (7.11), (7.4), (7.8), (7.14), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert [{{\mathcal {L}}}_n, \Pi _n^\bot ] {{\mathcal {L}}}_n^{- 1} \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0} \nonumber \\&\quad \lesssim \lambda ^{\theta } \,{\mathtt N}_{n}^{1-{\mathtt b}_{1}}\Big ( \Vert {\mathcal {L}}_{n}^{- 1} \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Vert {\mathcal {L}}_{n}^{- 1} \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + 1} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\gamma ^{-1} \,{\mathtt N}_{n}^{1- {\mathtt b}_{1}} \Big ( \Vert \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}+{\overline{\sigma }}} +{\mathtt N}_{n} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}+{\overline{\sigma }}} \Vert \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\overline{\sigma }}} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2- {\mathtt b}_{1}} \Big ( {\mathtt N}_{n}^{{\overline{\sigma }}} \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} +{\mathtt N}_{n}^{1+2{\overline{\sigma }}} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Vert {\mathcal {F}}(w_n) \Vert _{s_0} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2{\overline{\sigma }} +3- {\mathtt b}_{1}} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + C_* {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2{\overline{\sigma }} +3- {\mathtt b}_{1}} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Big ) , \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Vert [{{\mathcal {L}}}_n, \Pi _n^\bot ] {{\mathcal {L}}}_n^{- 1} \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} = \Vert [\Pi _n,{\mathcal {L}}_{n}] {{\mathcal {L}}}_n^{- 1} \Pi _n {{\mathcal {F}}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} \nonumber \\&\quad \lesssim \lambda ^{\theta } \,{\mathtt N}_{n} \Big ( \Vert {\mathcal {L}}_{n}^{- 1} \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}+1} \Vert {\mathcal {L}}_{n}^{- 1} \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + 1} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\gamma ^{-1} \,{\mathtt N}_{n} \Big ( \Vert \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}+{\overline{\sigma }}} +{\mathtt N}_{n} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}+{\overline{\sigma }}} \Vert \Pi _n {\mathcal {F}}(w_n) \Vert _{s_0 + {\overline{\sigma }}} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2} \Big ( {\mathtt N}_{n}^{{\overline{\sigma }}} \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} +{\mathtt N}_{n}^{1+2{\overline{\sigma }}} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Vert {\mathcal {F}}(w_n) \Vert _{s_0} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2{\overline{\sigma }} +3} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + C_* {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Big ) \nonumber \\&\quad \lesssim \lambda ^{\theta -1}\,{\mathtt N}_{n}^{2{\overline{\sigma }} +3} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0 + {\mathtt b}_{1}} + \Vert w_n \Vert _{s_0 + {\mathtt b}_{1}} \Big ). \end{aligned}$$\end{document}

By Lemma 3.1-(iii), (7.1), (7.5), (7.15) and (7.4), (7.16), (7.8), using also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\theta -1}=\lambda ^{\alpha -2+{\mathtt c}}<1$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert Q_n \Vert _{s_0}&\lesssim \lambda ^{\theta } {\mathtt N}_{n}^2 \Vert h_{n + 1} \Vert _{s_0}^2 \lesssim \lambda ^{\theta } {\mathtt N}_{n}^{2 {\overline{\sigma }} + 2} \lambda ^{- 2} \gamma ^{- 2} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0}^2 \\&\lesssim \lambda ^{\theta -2} {\mathtt N}_{n}^{2 {\overline{\sigma }} + 4} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0}^2 \lesssim {\mathtt N}_{n}^{2 {\overline{\sigma }} + 4} \lambda ^{- 1} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0}^2 , \\ \Vert Q_n \Vert _{s_0+{\mathtt b}_{1}}&\lesssim \lambda ^{\theta } \Vert h_{n+1}\Vert _{s_0+{\mathtt b}_{1}+1} \Vert h_{n+1} \Vert _{s_0+1} \lesssim \lambda ^{\theta } {\mathtt N}_{n}^{2} \Vert h_{n+1}\Vert _{s_0+{\mathtt b}_{1}} \Vert h_{n+1} \Vert _{s_0} \\&\lesssim \lambda ^{\theta -1}\gamma ^{-1} {\mathtt N}_{n}^{4{\overline{\sigma }}+3} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} + \Vert w_{n} \Vert _{s_0+{\mathtt b}_{1}} \Big ) \Vert {\mathcal {F}}(w_n) \Vert _{s_0} \\&\lesssim \lambda ^{\theta -1}\gamma ^{-1} {\mathtt N}_{n}^{4{\overline{\sigma }}+3} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} + \Vert w_{n} \Vert _{s_0+{\mathtt b}_{1}} \Big ) C_{*} {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \\&\lesssim \ {\mathtt N}_{n}^{4{\overline{\sigma }}+4} \Big ( \Vert {\mathcal {F}}(w_n) \Vert _{s_0+{\mathtt b}_{1}} + \Vert w_{n} \Vert _{s_0+{\mathtt b}_{1}} \Big ) . \end{aligned} \end{aligned}$$\end{document}

Therefore, by (7.17) and estimating (7.18) with (7.19), (7.20), (7.21), (7.22), we have proved the following inductive inequalities for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} and on the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_{n + 1}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\Vert w_{n + 1} \Vert _{s_0 + \mathtt b_1} + \Vert {\mathcal {F}}(w_{n+1})\Vert _{s_0+{\mathtt b}_{1}} \lesssim {\mathtt N}_{n}^{4 {\overline{\sigma }} + 4} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big ), \\&\quad \Vert {\mathcal {F}}(w_{n+1}) \Vert _{s_0} \lesssim {\mathtt N}_{n}^{2 {\overline{\sigma }} + 3 - \mathtt b_1} \Big ( \Vert {{\mathcal {F}}}(w_{n}) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big )\ + {\mathtt N}_{n}^{2 {\overline{\sigma }} + 4} \lambda ^{- 1} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0}^2. \end{aligned} \end{aligned}$$\end{document}

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1)_{n + 1}$$\end{document} . By (7.15), (7.8), (7.4) and (7.2), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert h_{n+1} \Vert _{s_0+{\overline{\sigma }}} \lesssim {\mathtt N}_{n}^{2{\overline{\sigma }}} {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \gamma ^{-1} \lambda ^{-1} \lesssim {\mathtt N}_{n}^{2{\overline{\sigma }}+1} {\mathtt N}_{n-1}^{-{\mathtt a}_{1}} \lambda ^{-1}, \end{aligned}$$\end{document}

which proves (7.6) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} . By (7.10), by the definition of the constants in (7.2) and by the smallness condition in (7.4), the estimate (7.5) at the step \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + 1$$\end{document} follows since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert w_{n + 1} \Vert _{s_0 + {\overline{\sigma }}} \leqslant \Vert w_0 \Vert _{s_0 + \overline{\sigma }} + \sum _{k = 1}^{n + 1} \Vert h_k \Vert _{s_0 + {\overline{\sigma }}} \leqslant \tfrac{1}{2} C_0 + C_*\sum _{k = 1}^\infty {\mathtt N}_{k-1}^{2 {\overline{\sigma }}} {\mathtt N}_{k -2}^{-\mathtt a_1} \lambda ^{- 1} \leqslant C_0, \end{aligned} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_{0}=\gamma ^{-1}=\lambda ^{\mathtt c}\gg 1$$\end{document} large enough and, if needed, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0>1$$\end{document} a bit larger. We prove now that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{n+1}$$\end{document} is a quasi-periodic traveling wave. By induction, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{n}$$\end{document} is a quasi-periodic. Moreover, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{n+1}$$\end{document} in (7.13) is a quasi-periodic traveling wave by Proposition 6.2, Lemma 3.1, Lemma 2.12, and the definition of the projections only on momentum preserving sites, see (2.6). Therefore, by (7.13), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{n+1}:=w_n + h_{n+1}$$\end{document} is a quasi-periodic traveling wave. Moreover, in the case where the forcing term f in (3.1) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi }, x)$$\end{document} , if by induction hypothesis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n = \textrm{odd}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}(w_n) = \textrm{even}({\varphi }, x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}_n^{- 1}$$\end{document} is reversibile, implying that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{n + 1} = \textrm{odd}({\varphi }, x)$$\end{document} and hence also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{n + 1} = w_n + h_{n + 1} = \textrm{odd}({\varphi }, x)$$\end{document} .

Proof of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}2)_{n + 1}, ({{\mathcal {P}}}3)_{n + 1}$$\end{document} . By the induction estimate (7.23) and using the induction hypothesis on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}2)_{n }, ({{\mathcal {P}}}3)_{n }$$\end{document} , we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert w_{n + 1} \Vert _{s_0 + \mathtt b_1} + \Vert {\mathcal {F}}(w_{n+1})\Vert _{s_0+{\mathtt b}_{1}}&\leqslant C\, {\mathtt N}_{n}^{4 {\overline{\sigma }} + 4} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big ) \\&\leqslant C C_* {\mathtt N}_{n}^{4{\overline{\sigma }} +4} {\mathtt N}_{n-1}^{\kappa } \leqslant C_* {\mathtt N}_{n}^{\kappa }, \end{aligned} \end{aligned}$$\end{document}

by the choice of the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} in (7.2) and taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_{0} \gg 1 $$\end{document} large enough. This latter chain of inequalities proves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}3)_{n + 1}$$\end{document} . Moreover, by (7.23), by the smallness condition (7.4) and by the choice of the constants in (7.2), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Vert {\mathcal {F}}(w_{n+1}) \Vert _{s_0}&\leqslant C\, {\mathtt N}_{n}^{2 {\overline{\sigma }} + 3 - \mathtt b_1} \Big ( \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0 + \mathtt b_1} + \Vert w_n \Vert _{s_0 + \mathtt b_1 } \Big )\ + C\,{\mathtt N}_{n}^{2 {\overline{\sigma }} + 4} \lambda ^{- 1} \Vert {{\mathcal {F}}}(w_n) \Vert _{s_0}^2 \\&\leqslant C C_* {\mathtt N}_{n}^{2 {\overline{\sigma }} + 3 - \mathtt b_1} {\mathtt N}_{n-1}^{\kappa } + C C_* {\mathtt N}_{n}^{2 {\overline{\sigma }} + 4} {\mathtt N}_{n-1}^{-2{\mathtt a}_{1}} \lambda ^{- 1} \\&\leqslant C_* {\mathtt N}_{n}^{-{\mathtt a}_{1}} . \end{aligned} \end{aligned}$$\end{document}

The proof of the claimed statement is then concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorem 1.3 and Measure Estimates

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \lambda ^{- \mathtt c}$$\end{document} as in (7.2), (7.4), in particular with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<{ \mathtt c} < {\overline{\tau }}\,^{-1}$$\end{document} . Then we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathtt N}_{0}^{{\overline{\tau }}} \lambda ^{- 1} = \gamma ^{- {\overline{\tau }}} \lambda ^{- 1} = \lambda ^{\mathtt c {\overline{\tau }} - 1} \ll 1 \end{aligned}$$\end{document}

by taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 0$$\end{document} large enough, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - \mathtt c \overline{\tau }> 0$$\end{document} . This implies that the smallness condition (7.4) is fullfilled. We define the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_\infty $$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {G}}}_\infty := \bigcap _{n \geqslant 0} {{\mathcal {G}}}_n \end{aligned}$$\end{document}

where the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n$$\end{document} are given in Proposition 7.3- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1)_n$$\end{document} . Hence, by Proposition 7.3- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}1)_n$$\end{document} , using a telescoping argument, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {\mathcal G}_\infty $$\end{document} , the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(w_n)_{n \geqslant 0}$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\infty \in H^{s_0 + {\overline{\sigma }}}_0$$\end{document} with respect to the norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )}$$\end{document} , and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert w_\infty \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \leqslant C_0, \quad \Vert w_\infty - w_n \Vert _{s_0 + {\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \lesssim {\mathtt N}_{n}^{2 {\overline{\sigma }}} {\mathtt N}_{n- 1}^{-\mathtt a_1} \lambda ^{- 1} \gamma ^{-1}, \quad \forall \,n \geqslant 1, \end{aligned}$$\end{document}

and consequently, by the same arguments, recalling (7.3),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \inf _{\omega \in {{\mathcal {G}}}_\infty }\Vert w_{\infty }(\cdot ; \omega ) \Vert _{s_0+{\overline{\sigma }}}&\geqslant \inf _{\omega \in {{\mathcal {G}}}_\infty } \Vert w_{0}(\cdot ; \omega ) \Vert _{s_0+{\overline{\sigma }}} - \Vert w_{\infty } -w_0 \Vert _{s_0+{\overline{\sigma }}}^{\textrm{Lip}(\gamma )} \\&\geqslant K \lambda ^{-{\mathtt c}} - C_* {\mathtt N}_{0}^{2{\overline{\sigma }}} \lambda ^{-1}\gamma ^{-1} \geqslant \tfrac{K}{2} \lambda ^{-{\mathtt c}} , \end{aligned} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough, by (7.2), (7.4) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\lambda ^{-{\mathtt c}}$$\end{document} . By (8.1) and Proposition 7.3- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\mathcal {P}}}2)_n$$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_\infty $$\end{document} , we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}(w_n) \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} . Therefore, the estimate (8.2) implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {F}}}(w_\infty ) = 0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_\infty $$\end{document} , whereas estimates (8.2)-(8.3) imply (1.12), after scaling back to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\lambda }:= \lambda ^{\theta } w_{\infty }=\lambda ^{\alpha -1+{\mathtt c}} w_{\infty }$$\end{document} (recall (4.5)). If we assume that f in (3.1) is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{even}({\varphi }, x)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_n = \textrm{odd}({\varphi }, x)$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} and hence also \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\infty = \textrm{odd}({\varphi }, x)$$\end{document} . The linearized equation at the quasi-periodic traveling wave solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v_\lambda (\lambda \, \omega t, x) = \lambda ^\theta w_\infty (\lambda \, \omega t, x)$$\end{document} , obtained by linearizing (1.7), has the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\partial _t h + L(\lambda \, \omega t)[h ] = 0, \quad L({\varphi }):= \textbf{a}_0({\varphi }, x) \cdot \nabla + {{\mathcal {E}}}_0({\varphi }) , \\&\quad \textbf{a}_{0}({\varphi }, x):= {\mathfrak {B}}\big [ v_\lambda ]({\varphi }, x) , \quad {\mathcal {E}}_{0}({\varphi })[h]:= \nabla v_\lambda ({\varphi }, x) \cdot {\mathfrak {B}}[h], \quad {\mathfrak {B}}= \nabla ^\perp (-\Delta )^{-1} . \end{aligned} \end{aligned}$$\end{document}

The linearized equation can be fully reduced to the constant coefficients equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _t \phi + \textbf{D}_\infty \phi = 0 \end{aligned}$$\end{document}

by the normal form scheme that we implemented in Sect. 5, see Propositions 5.5, 5.6, 5.13. In particular, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\infty = \textrm{odd}({\varphi }, x)$$\end{document} , we can apply Lemma 5.11 and we deduce that the eigenvalues of the reduced diagonal operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{D}_\infty $$\end{document} are purely imaginary. Consequently, we conclude that the quasi-periodic traveling wave solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_\lambda = \lambda ^\theta w_\infty $$\end{document} is linearly stable in the reversible case, and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \geqslant 0$$\end{document} , one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \phi (t) \Vert _{H^s_x} = \Vert \phi (0) \Vert _{H^s_x}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in {\mathbb {R}}$$\end{document} and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert h(t) \Vert _{H^s_x} \lesssim _s \Vert h(0 ) \Vert _{H^s_x}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in {\mathbb {R}}$$\end{document} .

By setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\lambda := {{\mathcal {G}}}_\infty $$\end{document} , the proof of Theorem 1.3 is concluded once we estimate the Lebesgue measure of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega {\setminus } \Omega _\lambda $$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^\nu $$\end{document} as in (1.11). This is the content of the last part of the paper.

The main result is the following proposition.

Proposition 8.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^\nu $$\end{document} be given as in (1.11) and let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau := \nu + 4. \end{aligned}$$\end{document}

Then, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega {\setminus } {{\mathcal {G}}}_\infty | \lesssim \gamma $$\end{document} . As a consequence, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \lambda ^{- \mathtt c}$$\end{document} as in (7.2), (7.4), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega {\setminus } {{\mathcal {G}}}_\infty | \lesssim \lambda ^{-{\mathtt c}}\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} .

The rest of this section is devoted to the proof of Proposition 8.1. By (8.1), (7.7), (3.6) and (3.5), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega \setminus {{\mathcal {G}}}_\infty \subseteq \big ( \Omega \setminus {\mathtt D}{\mathtt C}(\gamma ,\tau ) \big ) \bigcup \big ( {\mathtt D}{\mathtt C}(\gamma ,\tau ) \setminus \Omega _{\gamma } \big ) \bigcup _{n \geqslant 0} ({{\mathcal {G}}}_n \setminus {{\mathcal {G}}}_{n + 1}). \end{aligned}$$\end{document}

By Lemma 3.2 and the standard estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega \setminus {\mathtt D}{\mathtt C}(\gamma ,\tau )|$$\end{document} , it remains to estimate the measure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n \setminus {{\mathcal {G}}}_{n + 1}$$\end{document} . By (7.7) and using elementary properties of set theory, one has that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {G}}}_n \setminus {{\mathcal {G}}}_{n + 1} \subseteq ({\mathcal G}_n \setminus \Omega _\infty ^{\gamma _n}(w_n)) \cup ({{\mathcal {G}}}_n \setminus \Lambda _\infty ^{\gamma _n}(w_n))\, \end{aligned}$$\end{document}

where we recall that the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\infty ^{\gamma _n}(w_n)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _\infty ^{\gamma _n}(w_n)$$\end{document} are defined in (5.105), (6.3). Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_{n+1} \subseteq \mathcal {G}_{0} = \Omega _{\gamma }$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} , by (7.7).

Proposition 8.2

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} , the following estimates hold:

(i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_0 {\setminus } \Omega _\infty ^{\gamma _0}(w_0) \lesssim \gamma $$\end{document} and, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {G}}}_n {\setminus } \Omega _\infty ^{\gamma _n}(w_n)| \lesssim \gamma {\mathtt N}_{n}^{- (\tau - 1)}$$\end{document} ;

(ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_0 {\setminus } \Lambda _\infty ^{\gamma _0}(w_0) \lesssim \gamma $$\end{document} and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {G}}}_n {\setminus } \Lambda _\infty ^{\gamma _n}(w_n)| \lesssim \gamma {\mathtt N}_{n}^{- (\tau - 1)}$$\end{document} .

As a consequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {G}}}_0 {\setminus } {\mathcal G}_1| \lesssim \gamma $$\end{document} and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {G}}}_n {\setminus } {{\mathcal {G}}}_{n + 1}| \lesssim \gamma {\mathtt N}_{n }^{- (\tau - 1)}$$\end{document} .

Propositions 8.1 and 8.2 are proved at the end of this section. Now we estimate the measure of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n {\setminus } \Omega _\infty ^{\gamma _n}(w_n)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geqslant 0$$\end{document} . The estimate of the measure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n {\setminus } \Lambda _\infty ^{\gamma _n}(v_n)$$\end{document} can be done arguing similarly (it is actually even easier) and therefore it is omitted here. By the definitions (7.7), (5.105), we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{{\mathcal {G}}}_n \setminus \Omega _\infty ^{\gamma _n}(w_n) \subseteq \bigcup _{(\ell , j, j') \in {{\mathcal {I}}}}{{\mathcal {R}}}_{\ell j j'}(w_n), \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {I}}}:= \big \{ (\ell , j, j') \in ({\mathbb {Z}}^\nu \setminus \{0 \}) \times ({\mathbb {Z}}^2 \setminus \{ 0 \}) \times ({\mathbb {Z}}^2 \setminus \{ 0 \}): \pi ^\top (\ell ) + j - j' = 0 \big \},\nonumber \\ \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}}_{\ell j j'}(w_n):= \Big \{ \omega \in {{\mathcal {G}}}_n: |\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _\infty (j; w_n) - \mu _\infty (j'; w_n) | < \frac{2 \gamma _n\, \lambda }{\langle \ell \rangle ^\tau |j'|^\tau } \Big \}.\nonumber \\ \end{aligned}$$\end{document}

In the next lemma, we estimate the measure of the resonant sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n)$$\end{document} .

Lemma 8.3

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 0$$\end{document} , if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n) \ne \emptyset $$\end{document} , then we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {R}}}_{\ell j j'}(w_n)| \lesssim \gamma \,\langle \ell \rangle ^{-(\tau + 1)} |j'|^{-\tau }$$\end{document} .

Proof

Recalling (5.103), (5.109), we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _\infty (j;\omega ) \equiv \mu _\infty (j; \omega , w_n(\omega ))$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} as in (8.8), and we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \phi (\omega ):= \textrm{i}\,\lambda \,\omega \cdot \ell + \mu _\infty (j; \omega ) - \mu _\infty (j'; \omega ). \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \ne 0$$\end{document} , we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega = \frac{\ell }{|\ell |} s + v$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v \cdot \ell = 0$$\end{document} . We estimate the measure of the set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {{\mathcal {Q}}}&:= \Big \{ s: |\psi (s)| < \frac{2 \gamma _n\, \lambda }{\langle \ell \rangle ^\tau |j'|^\tau }, \ \ \frac{\ell }{|\ell |} \,s + v \in \Omega _{\gamma _n} \Big \} , \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \psi (s)&:= \phi \Big ( \tfrac{\ell }{|\ell |} s + v\Big ) = \textrm{i}\,\lambda |\ell | s + \mu _\infty (j; s) - \mu _\infty (j'; s) , \quad \mu _\infty (j; s) \equiv \mu _\infty \Big (j; \tfrac{\ell }{|\ell |} s + v \Big ). \end{aligned} \end{aligned}$$\end{document}

By Lemma 5.11, we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in {\mathbb {Z}}^2\setminus \{0\}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} | \mu _\infty (j; s_1) - \mu _\infty (j; s_2)| \lesssim \gamma ^{- 1}\lambda ^{\alpha - 1+{\mathtt c}} |j|^{- 1} |s_1 - s_2| , \end{aligned}$$\end{document}

and therefore

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} | \psi (s_1) - \psi (s_2)|&\geqslant \big ( \lambda |\ell | - C \lambda ^{\alpha - 1+{\mathtt c}} \gamma ^{- 1} \big ) |s_1 - s_2|\geqslant \frac{\lambda |\ell |}{2} |s_1 - s_2| , \end{aligned} \end{aligned}$$\end{document}

since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\alpha - 2+{\mathtt c}} \gamma ^{- 1} = \lambda ^{\alpha - 2(1-{\mathtt c})} \ll 1$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough (note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha - 2(1-{\mathtt c}) < 0$$\end{document} by (7.4)). This implies that the measure of the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {Q}}}$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{{\mathcal {Q}}}| \lesssim \frac{2 \lambda \gamma _n}{\langle \ell \rangle ^\tau |j'|^\tau } \frac{1}{\lambda |\ell |} \lesssim \frac{\gamma }{\langle \ell \rangle ^{\tau + 1} |j'|^{\tau }}. \end{aligned}$$\end{document}

By the Fubini theorem, we also get the claimed bound on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathcal R}_{\ell j j'}(w_n)|$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Several resonant sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{\ell j j'}(w_n)$$\end{document} are actually empty.

Lemma 8.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \geqslant {\mathtt N}_{n}^{\tau }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} . Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n) = \emptyset $$\end{document} .

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_n \subseteq {\mathtt D}{\mathtt C}(2\gamma _{n - 1},\tau )$$\end{document} , We have to prove that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < |\ell | \leqslant {\mathtt N}_{n}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \geqslant {\mathtt N}_{n}^\tau $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi ^\top (\ell ) + j - j' = 0$$\end{document} , then one has

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _\infty (j;w_n) - \mu _\infty (j';w_n)| \geqslant \frac{2\lambda \gamma _{n} }{{\langle {\ell }\rangle }^\tau } \geqslant \frac{2 \lambda \gamma _n}{\langle \ell \rangle ^\tau |j'|^\tau } , \end{aligned} \end{aligned}$$\end{document}

which implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n) = \emptyset $$\end{document} .

By Lemma 5.11, recalling also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =\alpha -1+{\mathtt c}$$\end{document} as in (4.5), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\textrm{i}\, \lambda \, \omega \cdot \ell + \mu _\infty (j;w_n) - \mu _\infty (j';w_n)| \geqslant \lambda |\omega \cdot \ell | - |\mu _{\infty }(j;w_n)| - |\mu _{\infty }(j';w_n)| \\&\quad \geqslant \frac{2\lambda \gamma _{n-1}}{{\langle {\ell }\rangle }^\tau } - \frac{C \lambda ^{\alpha -1+{\mathtt c}}}{\textrm{min}\{ |j|,|j'| \}} \\&\quad \geqslant \frac{2\lambda \gamma _{n}}{{\langle {\ell }\rangle }^\tau } + \frac{2\lambda \big (\gamma _{n - 1} - \gamma _{n} \big ) }{{\langle {\ell }\rangle }^\tau }- \frac{C \lambda ^{\alpha -1+{\mathtt c}}}{ {\mathtt N}_{n}^{\tau } } \geqslant \frac{2\lambda \gamma _{n}}{{\langle {\ell }\rangle }^\tau }, \end{aligned} \end{aligned}$$\end{document}

provided that, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{n - 1}-\gamma _{n} = \gamma \,2^{-n}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{2\lambda \big (\gamma _{n - 1} - \gamma _{n} \big ) }{{\langle {\ell }\rangle }^\tau }- \frac{C \lambda ^{\alpha -1+{\mathtt c}}}{ {\mathtt N}_{n}^{\tau } } \geqslant \frac{2\lambda \,\gamma \, 2^{-n}}{{\langle {\ell }\rangle }^{\tau }} \Big ( 1 - C \lambda ^{\alpha -2+{\mathtt c}} \gamma ^{-1} 2^{n-1} {\mathtt N}_{n}^{-\tau } {\langle {\ell }\rangle }^\tau \Big ) \geqslant 0 . \end{aligned}$$\end{document}

This latter condition is satisfied since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{\alpha - 2+{\mathtt c}} \gamma ^{- 1} = \lambda ^{\alpha - 2(1-{\mathtt c})} \ll 1$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} large enough (note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha - 2(1-{\mathtt c})< 0$$\end{document} by (7.4)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathtt N}_{0}\gg 1$$\end{document} large enough.

The claimed bound (8.10) is then proved. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

When non-empty, the sequence of resonant sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal {R}}_{\ell j j'}(w_n))_{n\in {\mathbb {N}}_0}$$\end{document} is nested.

Lemma 8.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \leqslant {\mathtt N}_{n}^\tau $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} . Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {R}}}_{\ell j j'}(w_n) \subseteq {{\mathcal {R}}}_{\ell j j'}(w_{n - 1}). \end{aligned}$$\end{document}

Proof

We split the proof in two steps.

Step 1. We claim that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j' \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_n$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}}|\mu _\infty (j; w_n) - \mu _\infty (j'; w_{n - 1})| \lesssim {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} + {\mathtt N}_{n - 1}^{- {\mathtt a}} \end{aligned}$$\end{document}

(the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt a$$\end{document} is defined in (5.56)). By (5.103) and the inductive definition of the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n$$\end{document} in (7.7), recalling (5.61), (5.105) and Lemma 5.12, we have the inclusion

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} {{\mathcal {G}}}_n \subseteq \Omega _n^{\gamma _{n - 1}}(w_{n - 1}) . \end{aligned} \end{aligned}$$\end{document}

We aim to apply Proposition 5.9- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)_n}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1 \equiv w_{n - 1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_2 \equiv w_n$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \equiv \gamma _{n-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \equiv \gamma _{n - 1} - \gamma _n = \gamma 2^{-n}$$\end{document} . By (7.6), we have (recall (5.58))

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&K {\mathtt N}_{n - 1}^{\tau + \tau ^2} \lambda ^{\alpha - 2+{\mathtt c}} \Vert w_n - w_{n - 1} \Vert _{s_0 + \Sigma ({\mathtt b})} \\&\quad \leqslant K C_{*} {\mathtt N}_{n - 1}^{\tau + \tau ^2 + 2 {\overline{\sigma }}} {\mathtt N}_{n-2}^{- \mathtt a_1} \, \lambda ^{\alpha - 2+{\mathtt c}} \lambda ^{- 1} \gamma ^{-1} \leqslant \gamma _{n - 1} - \gamma _n , \end{aligned} \end{aligned}$$\end{document}

for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K > 0$$\end{document} , where we have used that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{- 1} = {\mathtt N}_{0} \leqslant {\mathtt N}_{n - 1}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} and that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K C_{*} \, 2^n \, {\mathtt N}_{n - 1}^{\tau + \tau ^2 + 2 {\overline{\sigma }} + 2} {\mathtt N}_{n-2}^{- \mathtt a_1} \, \lambda ^{\alpha - 2+{\mathtt c}} \lambda ^{- 1} \leqslant 1 , \quad n \geqslant 1, \end{aligned}$$\end{document}

recalling that, by (5.56), (7.2) and the smallness condition (7.4), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\sigma }} > \Sigma ({\mathtt b})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt a_1 > \chi (\tau + \tau ^2 + 2 {\overline{\sigma }} + 2)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\tau }} \geqslant \tau + \tau ^2 + 2 {\overline{\sigma }} + 2$$\end{document} . The estimate (8.14) implies that Proposition 5.9- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S4)_n}$$\end{document} applies, leading, together with (8.13), to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n \subseteq \Omega _n^{\gamma _{n - 1}}(w_{n - 1}) \subseteq \Omega _n^{\gamma _{n}}(w_{n })$$\end{document} . Then, we apply the estimate (5.70) of Proposition 5.9- \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{(S3)_n}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_1 \equiv w_{n - 1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_2 \equiv w_n$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1 \equiv \gamma _n$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _2 \equiv \gamma _{n - 1}$$\end{document} , and obtain that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in \mathcal {G}_{n}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{j \in {\mathbb {Z}}^2 \setminus \{ 0 \}} |\mu _n(j; w_n) - \mu _n(j; w_{n - 1})|&\lesssim \lambda ^{\alpha - 1+{\mathtt c}} \Vert w_n - w_{n - 1} \Vert _{s_0 + {\overline{\sigma }}} \\&\lesssim {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} \lambda ^{\alpha - 2+{\mathtt c}}, \end{aligned} \end{aligned}$$\end{document}

where we have applied in the last inequality the estimate (7.6). Hence, by triangular inequality and by (5.104), (8.15), (7.5), we have, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {Z}}^2 {\setminus } \{ 0 \}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\mu _\infty (j; w_n) - \mu _\infty (j; w_{n - 1})| \leqslant |\mu _n(j;w_n) - \mu _n(j; w_{n - 1}) | \\&\quad +|\mu _\infty (j; w_n) - \mu _n(j; w_{n})| + | \mu _n(j; w_{n - 1}) - \mu _\infty (j; w_{n - 1}) | \\&\quad \lesssim {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} \lambda ^{\alpha - 2+{\mathtt c}} + 2 \lambda ^{-(M(\theta -1)+1)} \gamma ^{-(M+1)} {\mathtt N}_{n - 1}^{- \mathtt a} , \end{aligned} \end{aligned}$$\end{document}

which, by (7.4), implies the claimed estimate (8.12), since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M>\tfrac{1-{\mathtt c}}{2(1-{\mathtt c})-\alpha }$$\end{document} by (5.56), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha - 2+{\mathtt c}< 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \gg 1$$\end{document} is large enough.

Step 2. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_n$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \leqslant {\mathtt N}_{n}^\tau $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} . Note that, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|j - j'| \lesssim |\ell | \lesssim {\mathtt N}_{n}$$\end{document} and hence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{max}\{ |j|, |j'| \} \lesssim |j - j'| + \textrm{min}\{ |j|, |j'| \} \lesssim {\mathtt N}_{n} + {\mathtt N}_{n}^\tau \lesssim {\mathtt N}_{n}^\tau . \end{aligned}$$\end{document}

Therefore, by (8.12), using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega \in {{\mathcal {G}}}_n$$\end{document} , we get, for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C > 0$$\end{document} large enough,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&|\lambda \, \textrm{i}\, \omega \cdot \ell + \mu _\infty (j; w_n) - \mu _\infty (j'; w_n)| \\&\quad \geqslant |\lambda \,\textrm{i}\, \omega \cdot \ell + \mu _\infty (j; w_{n - 1}) - \mu _\infty (j'; w_{n - 1})| \\&\qquad - |\mu _\infty (j; w_n) - \mu _\infty (j; w_{n - 1})| - |\mu _\infty (j'; w_n) - \mu _\infty (j'; w_{n - 1})| \\&\quad \geqslant \frac{2 \lambda \gamma _{n - 1}}{\langle \ell \rangle ^\tau |j'|^\tau } - C \Big ( {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} + {\mathtt N}_{n - 1}^{- \mathtt a} \Big ) \geqslant \frac{2 \lambda \gamma _{n }}{\langle \ell \rangle ^\tau |j'|^\tau } , \end{aligned} \end{aligned}$$\end{document}

provided

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C \Big ( {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} + {\mathtt N}_{n - 1}^{- \mathtt a} \Big ) \leqslant \frac{2 \lambda (\gamma _{n - 1} - \gamma _n)}{\langle \ell \rangle ^\tau |j'|^\tau }. \end{aligned}$$\end{document}

Using that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} , the estimate (8.16), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{n - 1} - \gamma _n = \gamma 2^{- n}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{- 1} = {\mathtt N}_{0} \leqslant {\mathtt N}_{n}$$\end{document} , the latter condition is implied by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C' 2^{n - 1} {\mathtt N}_{n}^{\tau + \tau ^2 + 1} \lambda ^{- 1}\Big ( {\mathtt N}_{n-1}^{2 {\overline{\sigma }} + 1} {\mathtt N}_{n-2}^{-\mathtt a_1} + {\mathtt N}_{n - 1}^{- \mathtt a} \Big ) \leqslant 1, \quad n \geqslant 1, \end{aligned}$$\end{document}

for some constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C' \geqslant C$$\end{document} . The condition (8.18) is verified by (5.56), (7.2) and the smallness condition (7.4). We conclude then that the claimed inclusion (8.11) follows by the estimate (8.17). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 8.6

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} , the following inclusion holds:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathcal {G}}}_n \setminus \Omega _\infty ^{\gamma _n}(w_n) \subseteq \bigcup _{{\begin{array}{c}(\ell , j, j') \in {{\mathcal {I}}} \\ |\ell | \geqslant {\mathtt N}_{n} \end{array}}}{{\mathcal {R}}}_{\ell j j'}(w_n). \end{aligned}$$\end{document}

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} . By the definition (8.9), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n) \subseteq {{\mathcal {G}}}_n $$\end{document} . By Lemmata 8.4, 8.5, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\ell | \leqslant {\mathtt N}_{n}$$\end{document} , if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \geqslant {\mathtt N}_{n}^\tau $$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell , j j'}(w_n) = \emptyset $$\end{document} , and, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{min}\{ |j|, |j'| \} \leqslant {\mathtt N}_{n}^\tau $$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {R}}}_{\ell j j'}(w_n) \subseteq {{\mathcal {R}}}_{\ell j j'}(w_{n - 1})$$\end{document} . On the other hand, by (7.7), we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {G}}}_n \cap {{\mathcal {R}}}_{\ell j j'}(w_{n - 1}) = \emptyset $$\end{document} . Therefore (8.19) holds. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Proposition 8.2. We prove item (i). For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , j, j') \in {{\mathcal {I}}}$$\end{document} , one has that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^\top (\ell ) + j - j' = 0$$\end{document} and hence

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |j| \leqslant |\pi ^\top (\ell )| + |j'| \lesssim |\ell | + |j'| \lesssim \langle \ell \rangle |j'|. \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \geqslant 1$$\end{document} . By Lemmata  8.6, 8.3, the inclusion (8.19), (8.20) and (8.4), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \Big | {{\mathcal {G}}}_n \setminus \Omega _\infty ^{\gamma _n}(w_n)\Big |&\lesssim \sum _{(\ell , j, j') \in {{\mathcal {I}}} \atop |\ell | \geqslant {\mathtt N}_{n} } |{{\mathcal {R}}}_{\ell j j'}(w_n)| \lesssim \gamma \sum _{(\ell , j, j') \in {{\mathcal {I}}} \atop |\ell | \geqslant {\mathtt N}_{n} }\frac{1}{\langle \ell \rangle ^{\tau + 1} |j'|^\tau } \\&\lesssim \gamma \sum _{ |\ell | \geqslant {\mathtt N}_{n}} \frac{1}{\langle \ell \rangle ^{\tau - 1}} \sum _{j' \in {\mathbb {Z}}^2 \setminus \{ 0 \}} \frac{1}{|j'|^{\tau - 2}} \lesssim \gamma \, {\mathtt N}_{n}^{- (\tau - 1)}. \end{aligned} \end{aligned}$$\end{document}

Similarly, by the inclusion (8.7), Lemma 8.3 and using the standard volume estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega {\setminus } {\mathtt D}{\mathtt C}(\gamma , \tau )| \lesssim \gamma $$\end{document} , one also prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{{\mathcal {G}}}_0 {\setminus } \Omega _\infty ^{\gamma _0}(w_0)| \lesssim \gamma $$\end{document} . The item (ii) can be proved by similar arguments. Finally the estimate on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal G}_n {\setminus } {{\mathcal {G}}}_{n + 1}$$\end{document} follows by recalling formula (8.6) and by applying items (i), (ii).

Proof of Proposition 8.1

It follows by applying Proposition 8.2, Lemma 3.2, using the inclusion (8.5), the standard estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega \setminus {\mathtt D}{\mathtt C}(\gamma ,\tau ) |\lesssim \gamma $$\end{document} and the fact that the series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n \geqslant 1} {\mathtt N}_{n}^{-(\tau - 1)}$$\end{document} converges. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}