Convergence to normal forms of integrable PDEs

TL;DR

This paper establishes a method for transforming integrable PDEs like KdV, NLS, and Toda lattice into their normal forms using Poincaré normal form, enabling the construction of Birkhoff coordinates near the origin.

Contribution

It provides a novel iterative approach to explicitly construct local analytic transformations to normal forms for integrable PDEs in infinite-dimensional spaces.

Findings

Constructed local analytic coordinate transformations for integrable PDEs.

Applied the method to KdV, NLS, and Toda lattice equations.

Proved existence of Birkhoff coordinates near the origin.

Abstract

In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincar\'e normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincar\'e normal form and of the transformation to it, through a rapid convergence algorithm.

Equations

$${\Vert z\Vert }_{w^{(n)}}^2:=\sum _{j\in {\mathbb{Z}}^{\ast }}{w}_{|j|}^{(n)}|z_j{|}^2<\infty .$$

$$z^Q:=...z_{-k}^{q_{-k}}...z_{-1}^{q_{-1}}{z}_1^{q_1}....z_k^{q_k}...,|Q|:=\sum _{k\ge 1}\left(|q_{-k}|+|q_k|\right).$$

$$X(z)=\sum _{r\ge 0}\sum _{i,|Q|=r}{X}_{Q,i}{z}^Q\overrightarrow{e_i},$$

$$X(z)=\sum _{i,Q}{X}_{Q,i}{z}^Q\overrightarrow{e_i}.$$

$$Y:=(\exp U{)}^{\ast }X:=\sum _{k\ge 0}\frac{a{d}_U^k(X)}{k!},a{d}_U(X):=[U,X],$$

$$\underline{X}(z):=\sum _{Q,i}|X_{Q,i}|z^Q\overrightarrow{e_i},$$

$${\Vert \underline{X}\Vert }_r:=\underset{\Vert z\Vert \le r}{\sup }{\Vert \underline{X}(z)\Vert }_{+}.$$

$$\underline{{\bf F}}:=\sum_{i}\underline{F}^{i}\$$

$$F^i(z)=\sum _{Q,j}{F}_{Q,j}^i{z}^Q{\vec{e}}_j,$$

$$\underline{\mathbf{F}}(z)=\sum _{Q,j}\left(\sum _i\left|F_{Q,j}^i\right|\right)z^Q{\vec{e}}_j,$$

$$E^i(z):=z_i{\vec{e}}_i-z_{-i}{\vec{e}}_{-i}.$$

$$E^i=\sum _{j\in {\mathbb{Z}}^{\ast }}{\mu }_j^i{z}_j{\vec{e}}_j,{\mu }_j^i:={\delta }_j^i-{\delta }_{-j}^i,$$

$${\mathcal{N}}^{res}:=\{F\in \mathcal{N}|[E^i,F]=0,\forall i\}$$

$$[E^i,z^Q{\vec{e}}_j]=\left(\sum _{l\ge 1}{q}_l{\mu }_l^i-{\mu }_j^i\right)z^Q{\vec{e}}_j=:\left((Q,{\mu }^i)-{\mu }_j^i\right)z^Q{\vec{e}}_j.$$

$$F=F^{res}+F^{nres},F^{res}\in {\mathcal{N}}^{res},F^{nres}\in {\mathcal{N}}^{nres}.$$

$$d_0(U):={\left([E^i,U]\right)}_{i\ge 1}.$$

$$[E^i,F_d^j]=[E^j,F_d^i],i,j\ge 1.$$

$$F_d^j=\sum _{|Q|=d,i}{F}_{Q,i}^j{z}^Q{\vec{e}}_i.$$

$$\left((Q,{\mu }^i)-{\mu }_l^i\right)F_{Q,l}^j=\left((Q,{\mu }^j)-{\mu }_l^j\right)F_{Q,l}^i.$$

$$d_0(U)=\mathbf{F},$$

$$[E^j,U]=\sum _{Q,l}((Q,{\mu }^j)-{\mu }_l^j)U_{Q,l}{z}^Q{\vec{e}}_l=\sum _{Q,l}((Q,{\mu }^j)-{\mu }_l^j)\frac{F_{Q,l}^{i(Q,l)}}{(Q,{\mu }^{i(Q,l)})-{\mu }_l^{i(Q,l)}}{z}^Q{\vec{e}}_l$$

$$=\sum _{Q,l}{F}_{Q,l}^j{z}^Q{\vec{e}}_l.$$

$${\Vert \underline{U}\Vert }_r\le c{\Vert \underline{\mathbf{F}}\Vert }_r$$

$$(\exp U{)}_{\ast }{Z}^j=Z^j+[U,Z^j]+\frac{1}{2}[U,[U,Z^j]]+\cdots$$

$$[E^i,(\exp U{)}_{\ast }{Z}^j]=-[Z^j,[E^i,U]]+\frac{1}{2}[E^i,[U,[U,Z^j]]]+\cdots$$

$$(\exp U{)}_{\ast }{Z}^i=\sum _j(\sum _k\frac{U^k(a_{i,j})}{k!})E^j.$$

$$X^i=E^i+F^i,i\ge 1.$$

$${\mathcal{T}}_{\ast }{X}^i=E^i+N^i,\forall i\ge 1,$$

$$(X_{\mathcal{H}}{)}_k(z):=-i(\mathrm{sgn}k)\frac{\partial \mathcal{H}}{\partial z_{-k}}.$$

$$\{\mathcal{H},\mathcal{K}\}(z):=d\mathcal{H}(z)X_{\mathcal{K}}(z).$$

$${\mathcal{H}}^i={\mathcal{H}}_2^i+{\mathcal{K}}^i,{\mathcal{H}}_2^i:=z_i{z}_{-i},$$

Full text

Research of L. Stolovitch was supported by ANR-FWF grant ”ANR-14-CE34-0002-01” for the project “Dynamics and CR geometry” and by ANR grant “ANR-15-CE40-0001-03” for the project “BEKAM”

1. Introduction

In a Hilbert space $H$, consider a family $\{X^{i}\}$ of (germs of) analytic vector fields defined in a neighborhood of a common singular point, say the origin. We assume that they are pairwise commuting with respect to the Lie bracket. Consider the Taylor expansion $X^{i}=E^{i}+R^{i}$ of the fields at the origin, with $E^{i}$ the linear part. It is known since Poincaré, that each one of these vector fields can be transformed, by a formal change of variables $\hat{\mathcal{T}}_{i}$ into a Poincaré normal form $\hat{X}^{i}=(\hat{\mathcal{T}}_{i})_{*}X^{i}:=D\hat{\mathcal{T}}_{i}(\hat{\mathcal{T}}_{i}^{-1})X^{i}(\hat{\mathcal{T}}_{i}^{-1})$. By definition, it means that the Lie bracket $[E^{i},\hat{X}^{i}]=0$ vanishes. We then say that $\hat{\mathcal{T}}_{i}$ normalizes $X_{i}$. Since the family is abelian, i.e. $[X^{i},X^{j}]=0$ for al $i,j$, then one can show that there is a single $\hat{\mathcal{T}}$ that normalizes simultaneously the $X^{i}$’s in the sense that $[E^{i},\hat{\mathcal{T}}_{*}X^{j}]=0$, for all $i,j$.

In the same spirit, if $H$ is a symplectic space, one can study a family $\{{\mathcal{H}}^{i}={\mathcal{H}}^{i}_{2}+h.o.t\}$ of (germs of) analytic Hamiltonian functions which are higher order perturbations of quadratic Hamiltonians ${\mathcal{H}}^{i}_{2}$ and which are pairwise commuting with respect to the Poisson bracket associated to a symplectic form $\omega$. The normal forms of the Hamiltonians $\hat{\mathcal{T}}^{*}{\mathcal{H}}^{i}:={\mathcal{H}}^{i}\circ\hat{\mathcal{T}}$ are then called Birkhoff normal form. We have $\{\hat{\mathcal{T}}^{*}{\mathcal{H}}^{i},{\mathcal{H}}^{j}_{2}\}=0$ for all $i,j$ and $\hat{\mathcal{T}}$ is a formal symplectomorphism, i.e. $\hat{\mathcal{T}}_{*}\omega=\omega$.

A classical and fundamental problem in dynamics is to know under which assumption the normalizing transformation is not only formal, but also analytic. The motivation is to understand on the normal forms themselves many dynamical and geometrical properties which are not tractable directly on the original system. In finite dimension, this problem was solved by H. Rüssmann [Rüs67] for a single Hamiltonian vector field and by A.D. Brjuno [Bru72] for a single general germ of analytic vector field. In both cases, one assumes that ”the” formal normal form is of very special type, namely it has a very peculiar structure, nowaday called ”completely integrable”. For instance, in the Hamiltonian case, the formal Birkhoff normal form of the Hamiltonain ${\mathcal{H}}_{2}+h.o.t$ should be of the form $\hat{F}({\mathcal{H}}_{2})$, with a formal power series $\hat{F}(E)=E+h.o.t$ of the single variable $E$ (a different proof of the same result, avoiding superconvergence has been given in [LocatelliMele], by developing the methods of [LocGioZAMP]). Still in finite dimension, J. Vey proved two distinct results in the same spirit. On the one hand, he considered in [Vey78] a family of $n$ commuting Hamiltonian vector fields in $\mathbb{C}^{2n}$, whose linear parts are linearly independant. On the other hand, he considered in [Vey79] a family of $n-1$ commuting volume preserving vector fields in $\mathbb{C}^{n}$ whose linear parts are linearly independant. In both cases, he proved the existence of an analytic transformation to a normal form of the family near the origin. In the Hamiltonian case, H. Ito [Ito89, Ito92] impoved the results by essentially removing the condition of independance of the linear parts. N. T. Zung [Zun05, Zun02] generalized Vey’s Hamiltonian approach by considering $m$ ”linearly independent” vector fields having $n-m$ ”functionally independant” analytic first integrals in $\mathbb{C}^{n}$. He proved there the convergence of the transformation to normal forms. All these results have been unified in [Sto00, Sto05] (see also [Sto08]) in Rüssmann-Brjuno spirit : it is proved that if the formal normal form of the family has a very peculiar structure (called ”completely integrable”), and if the family of linear parts does not have ”bad small divisors”, then one can normalize analytically the family. One of the key points connecting the previous results with the later is that, preserving a structure such as a symplectic or a volume form, automatically implies that formal normal form of the family is ”completely integrable”.

The aim of this article is to devise such a normalizing scheme for ”complete sequences of integrable PDE’s in involution”. Algebraic ”Hierachies of PDEs” such as defined in [Mag78, Dic03] would have been the kind of objects we could have considered but their very algebraic nature does not seem to be suitable for our analysis. We consider sequences of integrable PDE’s such as a family of (germs of) analytic Hamiltonian functions $\{{\mathcal{H}}^{i}\}$ in a neighborhood of a common singular point of some suitable (infinite dimensional) Hilbert space $H$. We consider their associated Hamiltonian vector fields $\{X^{i}:=X_{{\mathcal{H}}^{i}}\}$ vanishing at a common singular point, say the origin. Since the Poisson bracket $\{{\mathcal{H}}^{i},{\mathcal{H}}^{j}\}=0$ for all $i,j$, then $[X^{i},X^{j}]=0$ for all $i,j$. As in finite dimension, the fact that all the $X^{i}$’s are symplectic implies that their formal normal form is of very special type, namely ”completely integrable” (see Definition 2.13 below). We shall show that the family of the linear parts $\{E^{i}\}$ at the origin, does not have ”small divisors” and prove, through a rapid convergence algorithm, that the transformation of the family to a normal form is convergent in a neighborhood of the origin.

We also prove that our algorithm allows to construct Birkhoff coordinates for all the vector fields $X^{i}$ and for all the vector fields commuting with each one of them. We recall that Birkhoff coordinates are a type of cartesian action angle coordinates $(x_{j},y_{j})$, s.t. all the Hamiltonians of the fields $X^{i}$ are funcition of $x_{j}^{2}+y_{j}^{2}$ only. We emphasize that our theorem is quite general and, as we will show, it applies to KdV, Toda and the defocusing NLS. Our starting point to address this problem is to consider the Lax pair [Lax68] $\frac{dL}{dt}=[B,L]$ associated to an Integrable Pde’s such as KdV. For instance, KdV equation on the circle, that is $\partial_{t}u-6u\partial_{x}u+\partial_{x}^{3}u=0$ for a function $u$ defined for $x$ on the circle $S^{1}$, is obtained for $L=\partial_{x}^{2}-u$, $B=-4\partial_{x}^{3}+6u\partial_{x}+\partial_{x}u$. It is known that the spectrum of $L$ is an invariant of the motion (i.e. independent of $t$) and, that the eingenvector equation turn out to be a Sturm-Liouville equation [Mar11]. It follows that the eigenvalues can be ordered as $\lambda_{0}<\lambda_{1}\leq\lambda_{2}<\cdots$. As shown in [GT84], the sequence of square of the gap lengths, $\{(\lambda_{2n}-\lambda_{2n-1})^{2}\}$ forms a family of analytic first integrals commuting pairwise for a suitable Poisson bracket. Our goal is be to transform analytically and simultaneously these Hamiltonians into a Birkhoff normal form.

We recall that a previous quite general theorem allowing to introduce Birkhoff coordinates is due to S. Kuksin and G. Perelman [KP10] who generalized Vey’s Hamiltonian approach to infinite dimension inspired by the scheme developed by H. Eliasson [Eli90]. In the present paper we show that Kuksin-Perelman’s result can also be deduced from our Theorem 2.1, in the sense that the assumptions of Kuksin-Perelman’s Theorem imply the assumptions of Theorem 2.1. Thus, in particular our main result applies to all the systems for which the assumptions of Kuksin-Perelman’s Theorem hold ([KP10, BM16, Mas18]).

We also recall that Birkhoff coordinates have been introduced originally in PDEs by Kappeler and coworkers [BBGK95, KP03, HK08, GK14, KLTZ09]. The idea of this series of papers is to consider the square of the spectral gaps associated to the Lax pair and to use them as a complete sequence of integrals of motion in order to apply Arnold Liouville procedure[Arn76] of construction of action angle variables (which of course has to be suitably generalized). Finally one regularizes the singularities of such variables by introducing cartesian type coordinates, which are the Birkhoff coordinates.

We emphasize that, althought Kuksin-Perelman’s and Kappeler’s approaches are different, they are intrinsically based on the symplectic structure and on Hamiltonian techniques.

In the present paper, we manage to directly normalize simultaneously the family of the first integrals by a Newton scheme (i.e. rapid convergence scheme such as for Nash-Moser theorem [BCP15, BBP10]). Furthermore, we emphasize that our scheme is finally unrelated to the symplectic geometry. As in finite dimension, symplectic geometry ensures that the formal Birkhoff normal forms of all the integrals are “completely integrable”, i.e. are of a very special form. Such a special form is crucial in order to estimate to solution of nonlinear cohomological equations.

In order to apply the algorithm in the present infinite dimensional context, we have to face several difficulties: the first one is to find a suitable norm to measure the size of a family of analytic vector fields, and the second one is the Lemma 3.2 which allow us to estimate the “nonlinear cohomological equation” without any small divisor problem. The last difficulty are located in Lemmas 3.4 and 3.5, which allow us to estimate the remainder and flows under the complete integrability assumption.

We also expect that our technique can be generalized to the case of systems preserving other structures, e.g. a volume form. Here we did not develop this because we are not aware of meaningfull examples to which such a theory would apply.

We recall that it is known how to put a system in normal form up to some reminder in a neighbourhood of a nonresonant fixed point (see e.g. [Bam03, BG06, BDGS07, Bam08]), however the technique we use here is completely different from the one of these papers, and we do not think that the ideas of those papers, applied to integrable PDEs could lead to the convergence result that we prove here.

Finally we remark that normal form results are often a fundemantal starting points for studying the stability of perturbed integrable PDEs (see e.g. [KP03, MP18, BKM18]).

Acknowledgements. We thank Michela Procesi for many discussions. We acknowledge the support of Università degli Studi di Milano and of GNFM.

2. Main results

2.1. Families of normaly analytic vector fields

Having fixed two sequences of weights $w^{(2)}_{j}\geq w^{(1)}_{j}>0$, $j\geq 1$, we define the Hilbert spaces $H=\ell^{2}_{w^{(1)}}$ and $H^{+}:=\ell^{2}_{w^{(2)}}$ where $\ell^{2}_{w^{(n)}}$ is the Hilbert spaces of the complex sequences $z:=\left\{z_{j}\right\}_{j\in{{\mathbb{Z}}^{*}}}$, ${{\mathbb{Z}}^{*}}:={\mathbb{Z}}-\{0\}$ s.t.

[TABLE]

In the following we will denote the norms simply by $\left\|z\right\|:=\left\|z\right\|_{w^{(1)}}$, and $\left\|z\right\|_{+}:=\left\|z\right\|_{w^{(2)}}$. Furthermore, we will denote by ${\bf e}:=\{\vec{e_{j}}\}_{j\in{{\mathbb{Z}}^{*}}}$ the vectors with components $(\vec{e}_{j})_{k}\equiv\delta_{j,k}$, the Kronecker symbol.

Let $Q\equiv(...,q_{-k},...,q_{-1},q_{1},...,q_{k},...)\in{\bf N}^{{{\mathbb{Z}}^{*}}}$ be an integer vector with finite support, then we write

[TABLE]

We shall denote ${\bf N}_{k}^{{{\mathbb{Z}}^{*}}}$ the set of $Q\in{\bf N}^{{{\mathbb{Z}}^{*}}}$ with $|Q|\geq k$.

A formal vector field $X$ is a formal sum of the form

[TABLE]

or simply

[TABLE]

Two formal vector fields will be said to be equal if the corresponding coefficients $X_{Q,i}$ coincide. A formal vector field $X$ is formally conjugate to a formal vector field $Y$, if there exists a formal vector vector field $U$ such that

[TABLE]

and $[.,.]$ is the commutator of vector fields.

Definition 2.1**.**

Let ${\mathcal{U}}\subset H$ be a neighborhood of the origin in $H$. A formal vector field $X$ as defined by (2.3) is said to be analytic from ${\mathcal{U}}$ to $H^{+}$ if the series (2.3) converges in $H^{+}$ uniformly for $z$ in ${\mathcal{U}}$. The space of such vector fields will be denoted by ${\mathcal{X}}^{\omega}({\mathcal{U}},H^{+})$. The space of germs at the origin of analytic vector fields with value in $H^{+}$ will be denoted by ${\mathcal{X}}^{\omega}_{0}(H,H^{+})$.

Let $X\in{\mathcal{X}}^{\omega}_{0}(H,H^{+})$ be a (germ of) analytic vector field at the origin of $H$ into $H^{+}$ and consider the vector field

[TABLE]

which in general is defined only on a dense subset of an open ball $H$.

Definition 2.2**.**

Let $X\in{\mathcal{X}}^{\omega}_{0}(H,H^{+})$ be an analytic vector field vanishing at the origin. We shall say that $X$ is normally analytic in a ball of radius $r$ if $\underline{X}$ is analytic in a ball of radius $r$ (in $H$ with values in $H^{+}$). In this case we will write $X\in{\mathcal{N}}_{r}$. We will write $X\in{\mathcal{N}}$ in all the cases where the value of $r$ is not important.

Remark 2.3**.**

The above definition immediately extends to the case of applications from $H$ to a general Banach space. In particular we will use it in Subsection 2.4 for the case where the target space is the space $B(H,H^{+})$ of bounded linear operators from $H$ to $H^{+}$.

In what follows, all analytic vector fields will be considered as defined in a neighborhood (precised or not) of the origin of $H$ with values in $H^{+}$.

A norm on ${\mathcal{N}}_{r}$ is given by

[TABLE]

Let $X,Y$ be normaly analytic vector fields. We shall say that $Y$ dominates $X$ and we shall write $X\prec Y$, if $|X_{Q,i}|\leq|Y_{Q,i}|$ for all indices.

Remark 2.4**.**

In particular, if $X\prec Y$, then $\left\|\underline{X}\right\|_{r}\leq\left\|\underline{Y}\right\|_{r}$ for any positive $r$.

Definition 2.5**.**

A family ${\bf F}=\{F^{i}\}_{i\geq 1}$ of normally analytic vector fields will be said to be summable if the vector field

[TABLE]

is normally analytic in a ball of radius $r$. In this case we will say that ${\bf F}\in{\mathcal{N\kern-3.0ptF}}_{r}$.

Remark 2.6**.**

Writing

[TABLE]

one has

[TABLE]

so that, for any $r>0$, $\left\|\underline{{\bf F}}\right\|_{r}$ bounds the norm of each one of the vector fields of the family, that is $\left\|\underline{F^{i}}\right\|_{r}\leq\left\|\underline{{\bf F}}\right\|_{r}$.

2.2. Normal forms

Consider the family ${\bf E}=\left\{E^{i}\right\}_{i\geq 1}$ of linear vector fields

[TABLE]

We will often use the notation

[TABLE]

which is ready for the generalization to the non Hamiltonian case.

Remark 2.7**.**

If the sequence $\frac{w_{j}^{(2)}}{w_{j}^{(1)}}\to\infty$, then the family ${\bf E}$ is not summably normally analytic according to our definition. Indeed, the vector field $\underline{{\bf E}}$ is the identity, which is not analytic as a map from $H$ to $H^{+}$.

Let ${\mathcal{N}}^{res}$ be the centralizer of the family ${\bf E}$, that is

[TABLE]

By the definition of ${\bf E}$, we have

[TABLE]

Hence, any function $F\in{{\mathcal{N}}}^{res}$ is obtained as the (possible infinite) linear combination of the monomials $z^{Q}\vec{e}_{j}$ for which $\left((Q,\mu^{i})-\mu_{j}^{i}\right)=0$ for all $i\geq 1$.

Let ${\mathcal{N}}^{nres}$ be the subspace of ${\mathcal{N}}$ generated by monomials $z^{Q}\vec{e}_{j}$ for which $\left((Q,\mu^{i})-\mu_{j}^{i}\right)\neq 0$ for some $i\geq 1$.

So, any vector field $F\in{\mathcal{N}}$ can be uniquely decomposed as

[TABLE]

A vector field $F\in{\mathcal{N}}^{res}$ will be called resonant, while a vector field $F\in{\mathcal{N}}^{nres}$ will be called non resonant. When speaking of the vector field $U$ which generates a coordinate transformation we shall say that it is normalized if $U^{res}=0$.

The same notation and terminology will be used also for families of vector fields, and in such a case we will write ${\mathcal{N\kern-3.0ptF}}^{nres}$ for a nonresonant family, namely a family composed by nonresonant vector fields and similarly for ${\mathcal{N\kern-3.0ptF}}^{res}$.

2.3. Cohomological equation

Let us consider the map $d_{0}$ which maps a homogeneous polynomial vector field $U$ of degree $d$ to the following family of homogeneous polynomial vector fields of degree $d$:

[TABLE]

This map is called the cohomological operator.

A family ${\bf F}=\{F^{i}_{d}\}_{i\geq 1}$ of homogeneous formal polynomial vector field of degree $d$ is called a cocycle with respect to the family ${\bf E}=\{E^{i}\}_{i\geq 1}$,if it satisfies:

[TABLE]

Let us write

[TABLE]

Therefore, equation (2.10) reads, for all $Q\in\mathbb{N}^{\mathbb{Z}^{*}}$ and $i,j\geq 1$ :

[TABLE]

As already pointed out, any cocycle ${\bf F}$ can be uniquely decomposed into a sum ${\bf F}={\bf F}^{res}+{\bf F}^{nres}$.

Lemma 2.8**.**

Let ${\bf F}=\{F^{i}\}_{i\geq 1}$ a formal homogeneous polynomial vector field of degree $d$ be a non resonant cocycle (i.e satisfying (2.10)). Then, it is a coboundary, that is there exists a formal homogeneous polynomial vector field $U$ of degree $d$ solution of the cohomological equation

[TABLE]

that is $[E^{i},U]=F^{i}$, for all $i$. Furthermore, there exists a unique normalized $U$ s.t. (2.12) holds.

Proof.

For each multiindex $Q\in\mathbb{N}^{\mathbb{Z}^{*}}_{2}$ and index $l\geq 1$ such that $(F^{j}_{Q,l})_{j}\neq 0$, there exists $i(Q,l)$ such that $(Q,\mu^{i})-\mu_{l}^{i}\neq 0$. Then set $U_{Q,l}:=\frac{F^{i(Q,l)}_{Q,l}}{(Q,\mu^{i(Q,l)})-\mu_{l}^{i(Q,l)}}$ and $U=\sum_{Q,i}U_{Q,i}z^{Q}\vec{e}_{i}$. Then according to (2.11), we have

[TABLE]

∎

Definition 2.9**.**

The family $\{E^{i}\}$ of linear vector field is said be small divisors free if there exists a positive constant $c$, such that for each $Q\in\mathbb{N}^{\mathbb{Z}^{*}}_{2}$ and $j\geq 1$, there is $i(Q,j)\geq 1$ such that $|(Q,\mu^{i(Q,j)})-\mu^{i(Q,j)}_{j}|>c^{-1}$.

Remark 2.10**.**

If the family ${\bf E}$ is small divisor free, then for any summable normally analytic cocycle ${\bf F}$, the unique normalized solution $U$ to (2.12) is normally analytic and satisfies, for some $r>0$

[TABLE]

for some constant $c$.

Definition 2.11**.**

A formal vector field $X$ (resp. a family ${\bf X}=\{X^{j}\}_{i\geq 1}$) is said be a normal form with respect to ${\bf E}$ if $[E^{i},{X}]=0$ for all $i\geq 1$ (resp. $[E^{i},X^{j}]=0$ for all $i,j\geq 1$).

Remark 2.12**.**

The above definition is taylor made for the case of Hamiltonian vector fields. In the case of vector fields preserving different structures the definition has to be modified following [Sto05].

Definition 2.13**.**

An analytic (resp. formal) normal form $X$ is said to be completely integrable if it can be written as $X=\sum_{j\geq 1}a_{j}E^{j}$ where $a_{j}$ are normally analytic (resp. formal) functions, invariants w.r.t ${\bf E}=\{E^{i}\}_{i\geq 1}$, i.e. $E^{i}(a_{j})=0$, for all $i,j\geq 1$.

Definition 2.14**.**

A family of formal vector fields is said to be formally completely integrable if it is formally conjugate to a completely integrable formal normal form.

Lemma 2.15**.**

A formal transformation of the form $\exp(U)$ with $U=\sum_{k\geq 2}U_{k}$, where $U_{k}$ is a homogeneous formal polynomial of degree $k$ commuting with each $E_{i}$, $i\geq 1$, conjugates a formal normal form of a family of formal vector fields to another normal form.

If one of the formal normal forms is completely integrable, so are all the other normal forms.

Proof.

First of all, if $\{Z^{j}\}$ and $\{(\exp U)_{*}Z^{j}\}$ are normal forms then $[U,E^{i}]=0$ for all $i$. Indeed, we have

[TABLE]

Taking the bracket with $E^{i}$ and using Jacobi identity, we obtain

[TABLE]

Let $U_{d_{0}}$ be lowest order term in the Taylor expansion of $U$ at origin. Then, one has $[E^{j},[E^{i},U_{d_{0}}]]=0$ for all $i,j$. Hence, $[E^{i},U_{d_{0}}]$ belongs to both the range and the kernel of the semi-simple map $[E_{i},.]$. Hence, $[E^{i},U_{d_{0}}]=0$ for all $i$. Hence, the lowest order term of $[Z^{j},[E^{i},U]]$ is $[E^{j},[E^{i},U_{d_{0}+1}]]$. On the other hand, since the bracket of resonant vector fields is still resonant, we have, for $k\geq 2$, $[E^{i},ad_{U}^{k}(Z^{j})]=[E^{i},ad_{U}^{k-1}([U-U_{d_{0}},Z^{j}])$ which is of order $\geq d_{0}+1+(k-1)(d_{0}-1)\geq 2d_{0}>d_{0}+1$. Hence, the lowest order term of $[E^{i},(\exp U)_{*}Z^{j}]$ is $[E^{j},[E^{i},U_{d_{0}+1}]]=0$ and we proceed by induction on the order.

Assume that the the family $\{Z^{j}\}$ is completely integrable. Transform it to another normal form ${\tilde{Z}^{j}}$ by a transformation $\exp U$. According to the first point, $U$ commutes with each $E^{i}$. Hence, it commutes with each $Z^{i}$ since $[U,\sum_{j}a_{i,j}E^{j}]=\sum_{j}a_{i,j}[U,E^{j}]+U(a_{i,j})E^{j}$. On the other hand, $E^{k}(U(a_{i,j}))=[E^{k},U](a_{i,j})=0$ for all $k$. So that

[TABLE]

∎

The main result of this paper is the following theorem.

Theorem 2.1**.**

Consider a family of analytic vector fields of the form

[TABLE]

Assume that

the family of linear vector fields ${\bf E}$ is small divisor free.

• 1.

${\bf F}\equiv\{F^{i}\}_{i\geq 1}$* is a summable family of normally analytic vector fields*

• 2.

there exists $c_{0}$ s.t. for $r_{0}$ small enough, one has $\left\|\underline{{\bf F}}\right\|_{r_{0}}\leq c_{0}r_{0}^{2}$

• 3.

$[X^{i};X^{j}]\equiv 0$, $\forall i,j$.

• 4.

${\bf X}$* is formally completely integrable.*

Then there exist constants $r_{*}>0$, $c_{2},c_{3}$, a neighborhood ${\mathcal{U}}\supset B_{r_{*}}$ of the origin and an analytic coordinate transformation ${\mathcal{T}}:{\mathcal{U}}\to H$ s.t.

[TABLE]

where ${\mathcal{N\kern-3.0ptF}}_{r_{*}}\ni{\bf N}\equiv\left\{N^{i}\right\}_{i\geq 1}$ is a completely integrable normal form.

Furthermore, $\forall r<r_{*}$ the following estimates hold:

• i.

$\left\|{\bf N}\right\|_{r}\leq c_{2}r^{2}$,

• ii.

$\sup_{\left\|z\right\|\leq r}\left\|z-{\mathcal{T}}(z)\right\|_{+}\leq c_{3}r^{2}$.

Remark 2.16**.**

From the proof it is clear that if one endows the Hilbert space by the symplectic structure ${\rm i}dz_{-k}\wedge dz_{k}$ and the vector fields $F^{i}$ are Hamiltonian for any $i$, then the transformation ${\mathcal{T}}$ is canonical. Here we did not assume the fields $X^{i}$ to be Hamiltonian. In the Hamiltonian case Assumption 4 would be automatic.

We expect that the result can be extended also to other preserved structure, like volume in phase space, but the present proof rely on the structure of the family ${\bf E}$.

Remark 2.17**.**

The Hilbert spaces considered can chosen to be more general. For instance, it could be be spaces of sequences indexed over $\mathbb{Z}^{d}\setminus\{0\}$.

We are now going to give a more precise statement for the Hamiltonian case, showing in particular that the transformation ${\mathcal{T}}$ introduces Birkhoff coordinates for the integrable Hierarchy associated to the fields $\{X^{i}\}$.

Thus, in the space $H$, we introduce the symplectic form ${\rm i}dz_{-k}\wedge dz_{k}$. Given an analytic function ${\mathcal{H}}\in C^{\omega}(H,\mathbb{R})$, we define the corresponding Hamiltonian vector field $X_{{\mathcal{H}}}$ as the vector field with $k$-th component

[TABLE]

Given also a second function ${\mathcal{K}}\in C^{\omega}(H,\mathbb{R})$ we define their Poisson Bracket by

[TABLE]

It is well known that such a quantity can fail to be well defined, nevertheless in all the cases we will consider it will be well defined.

Consider now a sequence of analytic Hamiltonians ${\mathcal{H}}^{i}$ of the form

[TABLE]

and ${\mathcal{K}}^{i}$ having a zero of order at least $3$ at the origin.

Corollary 1**.**

Assume that the vector fields $X^{i}:=X_{{\mathcal{H}}^{i}}$ fulfill the assumtpions of Theorem 2.1, then the coordinate transformation ${\mathcal{T}}$ is canonical. Furthermore, given any analytic Hamiltonian ${\mathcal{H}}$ with a zero of order 2 at the orgin, such that

[TABLE]

one has that ${\mathcal{H}}\circ{\mathcal{T}}^{{-1}}$ is a function of $\{(z_{j}z_{-j})\}_{j\geq 1}$ only.

Proof.

The proof follows the proof of Corollary 2.13 of [BM16]. First, it is clear that $E^{j}$ is the Hamiltonian vector field of ${\mathcal{H}}^{j}_{2}$. Denote $\tilde{\mathcal{H}}:={\mathcal{H}}\circ{\mathcal{T}}^{{-1}}$, thus, from the property that ${\mathcal{T}}_{*}X^{j}$ is in normal form one has that

[TABLE]

from which $\{\tilde{\mathcal{H}},{\mathcal{H}}^{j}_{2}\}=c^{j}$. However, since both $\tilde{\mathcal{H}}$ and ${\mathcal{H}}_{2}$ have a zero of order 2 at the orgin, the constants must vanish. Expand now $\tilde{\mathcal{H}}$ in Taylor series, one has

[TABLE]

where we denoted $z_{+}:=\{z_{j}\}_{j\geq 1}$ and $z_{-}:=\{z_{-j}\}_{j\geq 1}$. Then equation (2.16) implies that in each term of the summation $\alpha=\beta$, therefore $\tilde{\mathcal{H}}$ is a function of $z_{j}z_{-j}$ only. ∎

2.4. Kuksin-Perelman’s Theorem

In this section we recall the Vey type theorem obtained by Kuksin and Perelman in [KP10] (see also [BM16, Mas18]) and prove that it can be obtained as a corollary of Theorem 2.1.

We come to the assumptions of the Kuksin-Perelman’s Theorem.

Consider an analytic map $\Psi$ of the form

[TABLE]

with $G\in{\mathcal{N}}_{R}$ (with some $R>0$) having a zero of second order at the origin. For $j>0$ consider also the functions $I_{j}(z):=\Psi_{j}(z)\Psi_{-j}(z)$ and the Hamiltonian vector fields $X^{j}:=X_{I_{j}}$.

Assume that the following Hypotheses hold:

• (KP1)

The functions $I_{j}(z)$ pairwise commute, namely $\left\{I_{j};I_{k}\right\}\equiv 0$ forall $j,k\geq 1$.

• (KP2)

the maps $\underline{dG}$ and $\underline{dG^{*}}$ are analytic as maps from $B_{R}$ to $B(H,H^{+})$.

Theorem 2.2** (Kuksin-Perelman).**

Assume that (KP1) and (KP2) hold, then the same conclusions of Theorem 2.1 and Corollary 1 hold.

Proof.

It is enough to show that the assumptions (KP1-KP2) imply the assumptions of Theorem 2.1 with the fields $X^{j}:=X_{I_{j}}$. First remark that assumption 3 of Theorem 2.1 follows from (KP1), while assumption 4 follows from the fact that the fields $X^{i}$ are Hamiltonian. Assumption 0 follows from the structure (2.17) of the function $\Psi$.

In order to verify assumptions 1, compute explicitely the components of the vector fields $X_{I_{l}}$. For $k\geq 1$ its $k-th$ component is given by

[TABLE]

the first term contribute to $E^{l}$, while all the other ones contribute to $F^{l}$. From (2.18) we have that the $k$-th component of $\underline{{\bf F}}$, ($k\geq 1$) is given by

[TABLE]

We have to show that each one of the terms of this expression define the $k$-th component of an analytic vector field. For the first term this is a trivial consequence of the fact that $G\in{\mathcal{N}}$. Consider the second term. In order to see that it is analytic we write it in terms of $\underline{dG^{*}}$. To this end define the involution $(Iz)_{k}:=z_{-k}$ and the truncation operator $(Tz)_{k}=z_{k}$ if $k\geq 1$ and zero otherwise. Then the second term of the above expression is the $-k$-th component of $\underline{dG^{*}}(ITz)$, which belongs to ${\mathcal{N}}$ by assumption (KP1). All the other terms can be dealt with in the same way geting that assumption 1 is fulfilled. Assumption 2 is a direct consequence of the fact that $G$ has a zero of order 2 at the origin. ∎

3. Proof of the main theorem

3.1. Nonlinear cohomological equation

Assume the abelian family ${\bf X}=\{X^{i}\}$ is normalized up to order $m=2^{k}$:

[TABLE]

where $\textbf{N}_{\leq m}\in{\mathcal{N\kern-3.0ptF}}_{R_{m}}$ is a completely integrable normal form of degree $m$; we shall write $NF^{i}_{\leq m}=E^{i}+N^{i}_{\leq m}=\sum_{j\geq 1}(\delta_{i,j}+a_{i,j})E^{j}$ where $a_{i,j}$ are polynomials of degree $\leq m-1$ that are common first integrals of $\bf E$. Let us Taylor expand $R^{i}_{\geq m+1}=B^{i}_{\leq 2m}+\tilde{R}^{i}_{\geq 2m+1}$ up to degree $2m$. We shall (mostly) omit the dependence on $m$ in this section. Since $X^{i}$ and $X^{j}$ are pairwise commuting, then

[TABLE]

Therefore, the truncation at degree $\leq 2m$ gives

[TABLE]

where $J^{2m}(V)$ denotes the $2m$-jet of $V$.

Lemma 3.1**.**

Let ${\bf B}$ be a nonresonant family and ${\bf N}$ a completely integrable normal form. Assume that they fulfill (3.20). Then there exists a unique $U$ normalized (i.e. no resonant term in expansion) such that for all $j$ one has $J^{2m}([NF^{j},U])=B^{j}$.

Proof.

We give here a direct proof although a more conceptual proof involving spectral sequences can be found in [Sto00][proposition 7.1.1] for the finite dimensional case. For any integer $m+1\leq k\leq 2m$, the homogenous polynomial of degree $k$ of eq.(3.20) is

[TABLE]

Let us prove, by induction on the integer $m+1\leq k\leq 2m$, that there exists a unique normalized polynomial $V_{k}$ homogeneous of degree $k$, such that

[TABLE]

that is $J^{k}\left([NF^{i},\sum_{p=m+1}^{k}V_{p}]\right)=J^{k}(B^{i})$.

For $k=m+1$, the equation $(\ref{nonlin-compt})$ leads to $[E^{i},Z^{j}_{m+1}]=[E^{j},Z^{i}_{m+1}]$. According to the Lemma 2.8, there exists a unique normalized $V_{m+1}$ homogeneous of degree $m+1$ such that, for all $1\leq i$, $[E^{i},V_{m+1}]=Z^{i}_{m+1}$.

Let us assume that the result holds for all integers $q<k$. Let $2\leq p\leq k-m$ be an integer, then $m+1\leq k-p+1<k$. Let us first recall that, by assumptions, $[NF_{k}^{i},NF_{k^{\prime}}^{j}]=0$ for all integers $1\leq i,j$ and $1\leq k,k^{\prime}$.

Thus, by Jacobi Identity, we have

[TABLE]

With these remarks as well as $(\ref{result-rec})$, it follows, by induction, that

[TABLE]

Since $[NF^{j}_{q},[NF^{i}_{p},V]]=[NF^{i}_{p},[NF^{j}_{q},V]]$, then exchanging $j$ and $i$ leads to

[TABLE]

Summing over $2\leq p\leq k$ and using the compatibility condition $(\ref{nonlin-compt})$ leads to

[TABLE]

But, the same argument as in the proof of the first point of this proposition will show that, $\{\sum_{p=2}^{k-m}[V_{k-p+1},NF_{p}^{i}]\}$ is a non-resonant family of homogenoues vector fileds of degree $k$. Therefore, according to Lemma 2.8, there exists a unique normalized $V_{k}$ such that, for all $i\geq 1$,

[TABLE]

which ends the proof of the induction and the proposition.

∎

Let us construct and estimate the unique nonresonant solution $U$ (i.e. with $U^{res}\equiv 0$), of order $\geq m+1$ and degree $\leq 2m$ of the nonlinear cohomological equation, namely

[TABLE]

where $B^{i}_{nres}$ denotes the nonresonant projection of $B^{i}$.

Lemma 3.2**.**

Assume that, for all $\|z\|\leq r$, for all $v\in H$

[TABLE]

Then (3.23) has a unique nonresonant solution $U$ which satisfies

[TABLE]

Proof.

Let us write (3.23) as

[TABLE]

where $Z^{i}_{\geq 2m+1}:=J^{2m}[NF^{i},U]-[NF^{i},U]$.

Let $\lambda_{i}^{(d)}$ be an eigenvalue of the operator $[E^{i},\cdot]$ in the space of formal homogeneous polynomial vector fields of degree $d$. Let $h_{i,\lambda_{i}^{(d)}}$ be the associated eigenspace.

Remark 3.3**.**

Due to the definition of the family ${\bf E}$, we have $\lambda_{i}^{(d)}=q_{i}-q_{-i}-s_{i}$ where $q_{i}$ denotes the $i$th component of a multiindex $Q=(\cdots q_{-i-1},q_{-i},\cdots,q_{-1},q_{1},\cdots q_{i},q_{i+1},\cdots)$ with modulus $d$ and $s_{i}$ is 1,-1 or 0. Indeed, we have $[E^{i},x^{Q}\vec{e}_{k}]=(q_{i}-q_{-i}-s_{i})x^{Q}\vec{e}_{k}$ with $s_{i}=1$ if $k=i$, $s_{i}=-1$ if $k=-i$ and $s_{i}=0$ otherwise.

Let $\lambda^{(d)}=(\lambda_{1}^{(d)},\lambda_{2}^{(d)},\ldots)$ be a collection of such eigenvalues. We shall say that $\lambda^{(d)}$ is a generalized eigenvalue of degree $d$. If $\lambda^{(d)}\neq 0$, then only a finite number of its components are non zero. Let us denote $\text{Supp}(\lambda^{(d)})$, the support of $\lambda^{(d)}$, that is the set of indexes $j$ such that $\lambda^{(d)}_{j}\neq 0$. From now on, we shall write $\lambda$ for $\lambda^{(d)}$, if there is no confusion.

We remark that, given $U\in\cap_{i\geq 1}h_{i,\lambda_{i}^{(d)}}$, and any function $a$ which is a common first integral of the family $E^{i}$, namely s.t. $E^{k}(a)=0$, $\forall k$, one has

[TABLE]

thus it is convenient to denote

[TABLE]

We now show that $[NF^{i},.]$ leaves invariant $H_{\lambda}$ (where we omitted the index $d$ from $\lambda$). We have

[TABLE]

Here, $U(a_{i,j})$ denotes the Lie derivative of $a_{i,j}$ along $U$. Since the $E^{i}$’s are pairwise commuting and since the $a_{i,j}$’s are first integrals of $\bf E$, we have

[TABLE]

On the other hand, we have

[TABLE]

From which the invariance of $H_{\lambda}$ follows.

Let $U_{\lambda}$ (resp. $F^{i}_{\lambda}$) be the projection onto $H_{\lambda}$ of $U$ (resp. $F^{i}$). Therefore, the projection onto $H_{\lambda}$ of equation (3.24) reads

[TABLE]

Using (3.26), this equation reads

[TABLE]

Let $\epsilon_{i}$ be the sign of $\lambda_{i}$, if $i\in\text{Supp}(\lambda)$. Let us multiply the $i$th-equation by $\epsilon_{i}$ and then let us sum up over $i\in\text{Supp}(\lambda)$. We obtain

[TABLE]

Let us define

[TABLE]

Remark that it is an analytic function whose value at [math] is $|\lambda|$; furthermore one has $E^{j}(b_{\lambda})=0$, $\forall j$. Let us consider the operator

[TABLE]

We have $P_{\lambda}^{2}=0$. Indeed, since the $a_{i,j}$ are first integrals of $\bf E$, we have

[TABLE]

Similarly one has $P_{\lambda}(P_{\lambda}(./b_{\lambda}))=0$. As a consequence, the nonresonant solution of equation (3.28) is

[TABLE]

Summing up over the set of generalized eigenvalues $\lambda$ of degree $m+1\leq d\leq 2m$, and applying $J^{2m}$ we obtain

[TABLE]

Since $U$ is of degree $\leq 2m$, we can substitute $B_{\lambda}$ to $F_{\lambda}$, thus we are led to the final definition of $U$, namely

[TABLE]

where $\tilde{B}_{\lambda}:=\sum_{i\in\text{Supp}(\lambda)}\epsilon_{i}B^{i}_{\lambda}$. We now estimate such a quantity. Remark first that one has

[TABLE]

so that we have

[TABLE]

On the other hand, given an orthonormal basis ${\bf e}$ of $H^{+}$, a sequence $\{G_{\lambda}\}$ of vectors with nonnegative coordinates on ${\bf e}$ and a bounded sequence $\{g_{i}\}$ of nonnegative numbers, we have

[TABLE]

Evaluating at a point near the origin in the domain, we can apply this with $g_{\lambda}=\frac{1}{|\lambda|-\underline{c_{\lambda}}}$ and $G_{\lambda}=\sum_{i\in\text{Supp}(\lambda)}\underline{B^{i}_{\lambda}}$ Hence, we obtain

[TABLE]

In order to estimate $\underline{c_{\lambda}}$, remark first that according to (3.29), we have

[TABLE]

To estimate $\beta_{j}:=\sum_{i}\underline{a_{ij}}$, we proceed as follows. According to (2.9), we have $\underline{N^{i}}=\sum_{j\in\mathbb{Z}^{*}}\underline{a_{i,j}}z_{j}\vec{e}_{j}$ so that $\underline{{\bf N}}=\sum_{j\in\mathbb{Z}^{*}}\left(\sum_{i}\underline{a_{i,j}}\right)z_{j}\vec{e}_{j}=\sum_{j}\beta_{j}z_{j}\vec{e}_{j}$. Hence, we have

[TABLE]

Since the previous equality involves only vectors with nonnegative coefficients, we have

[TABLE]

So, $\forall v\in H$ and for all $\|z\|\leq r$, we have

[TABLE]

Taking $v:=v_{k}\vec{e}_{k}=1/\sqrt{w_{k}^{(1)}}\vec{e}_{k}$, which has norm 1, one gets

[TABLE]

Inserting in (3.34) one gets

[TABLE]

hence

[TABLE]

Since the familly ${\bf E}$ is small divisor free, then we always have $1\leq|\lambda|$ (we have set $c=1$ for simplicity), then by (3.33)

[TABLE]

as soon as $\|\underline{{\bf B}}\|_{+}\leq\epsilon$. On the other hand, we have

[TABLE]

According to (3.36), we have

[TABLE]

As in (3.33), we have

[TABLE]

Hence, for $\|z\|\leq r$,

[TABLE]

Collecting estimates (3.37) and (3.38), we obtain

[TABLE]

and remarking that, for functions of class ${\mathcal{N}}$ the projector $J^{2m}$ does not increase the norm, one gets

[TABLE]

∎

3.2. Flow of normally analytic vector fields

In this section we study the flow $\Phi^{t}$ of a vector field $U\in{\mathcal{N}}_{r}$. In particular we will prove the following Lemma

Lemma 3.4**.**

Assume that $U\in{\mathcal{N}}_{r}$ for some $r>0$ fulfills $\epsilon:=\left\|U\right\|_{r}<\frac{\delta}{4e}$ and let ${\bf F}\in{\mathcal{N\kern-3.0ptF}}_{r}$ and $\delta<r$. Then the family $(\Phi^{-1})^{*}{\bf F}\equiv\{(\Phi^{-1})^{*}F^{i}\}_{i\geq 1}$ is summable normally analytic and, defining $S^{i}:=(\Phi^{-1})^{*}F^{i}-F^{i}$ and $\tilde{S}^{i}:=(\Phi^{-1})^{*}F^{i}-F^{i}-[U,F^{i}]$, one has

[TABLE]

Proof.

To start with, we remark that, since $\sup_{\left\|z\right\|<r}\left\|U(z)\right\|_{+}\leq\left\|\underline{U}\right\|_{r}$, $\forall\left|t\right|\leq 1$, one has

[TABLE]

and therefore $z\in B_{r-\delta}$ implies $\Phi^{t}(z)\in B_{r}$ i.e. $\Phi^{t}(B_{r-\delta})\subset B_{r}$ ($B_{r}$ denoting the ball in $H$ of radius $r$ centered at zero). Thus the flow is well defined and analytic at least up to $|t|=1$. By Taylor expanding in $t$ at $t=0$, one has

[TABLE]

where $Ad_{U}G:=[U,G]$. To estimate this family remark first that

[TABLE]

Summing over $i$ one gets

[TABLE]

and, by induction on $k$

[TABLE]

Thus we have

[TABLE]

In order to estimate the r.h.s. remark first that, for any family ${\bf G}\in{\mathcal{N\kern-3.0ptF}}_{r-\delta-\delta_{1}}$ (for some $\delta,\delta_{1}\geq 0$), we have, by Cauchy estimate

[TABLE]

Fix now some $k\geq 0$, define $\delta^{\prime}:=\delta/k$ and look for constants $C^{(k)}_{l}$, $0\leq l\leq k$ s.t.

[TABLE]

Of course, by (3.43) they can be recursively defined by

[TABLE]

which gives

[TABLE]

taking $l=k$ this produces an estimate of the general term of the r.h.s. of (3.42):

[TABLE]

where we used $k!\geq k^{k}e^{-k+1}$. Summing over $k\geq 1$ or $k\geq 2$, one gets the thesis. ∎

Although the family ${\bf E}$ is not summably normally analytic, its composition with the flow has the following remarkable property.

Lemma 3.5**.**

Assume that $U\in{\mathcal{N}}_{r}$ for some $r>0$ fulfils $\epsilon:=\left\|U\right\|_{r}<\frac{\delta}{8e}$ with $0<\delta<r$; then the family ${\bf T}\equiv\{(\Phi^{-1})^{*}E^{i}-E^{i}-[U,E^{i}]\}_{i\geq 1}$ is summably normally analytic and one has

[TABLE]

Proof.

We proceed as in the proof of the previous Lemma except that we compute explicitly the first term of the expansion (3.41).

One has $D\underline{U}\,\underline{{\bf E}}=(D\underline{U}(z))z$ and $\sum_{i}(D\underline{E^{i}})\underline{U}=\underline{U}$, so we get (for any $\delta^{\prime}<r$),

[TABLE]

So, by (3.44),

[TABLE]

thus

[TABLE]

Taking $\delta^{\prime}=\delta/2$ one gets the thesis.∎

3.3. Iteration

We use $U$ to generate a change of variables which is the time 1 flow, $\Phi$ of the system $\dot{z}=U(z)$. We have

[TABLE]

By assumption, $B_{0}^{i}=\sum_{j\geq 1}\tilde{a}_{i,j}E^{j}$ where $\tilde{a}_{i,j}$ are polynomials of degree $\leq 2m-1$ that are common first integrals of $\bf E$.

By assumption, we have ${\bf X}={\bf E}+{\bf F}$ and there exists $c_{0}$ such that

[TABLE]

for some small parameter $r_{0}$. We also fix two large constants $c_{1}$ and $b{\geq 1}$ (we will track the dependence of everything on such constants). Their precise value will be decided along the procedure.

We denote $m:=2^{k}$, $k\geq 0$ then the sequences we are interested in are defined by

[TABLE]

In the appendix we will prove that the following properties hold

[TABLE]

provided $c_{1}\geq 4^{b+2}/3$. Actually we take

[TABLE]

We will also prove that

[TABLE]

Consider the following inequlities (with $m=2^{k}$)

[TABLE]

Lemma 3.6**.**

Assume

[TABLE]

take

[TABLE]

Assume that the inequalities (3.59)-(3.61) hold with some $k\geq 0$. Let $\Phi_{m}$ be the flow generated by $U_{m}$ defined by (3.31). It conjugates the family $X^{i}_{m}=E^{i}+N^{i}_{\leq m}+R^{i}_{\geq m+1}$ to the family $X^{i}_{2m}:=E^{i}+N^{i}_{\leq 2m}+R^{i}_{\geq 2m+1}$ and (3.59), (3.60), (3.61) hold for the new $N$ and $R$ with $k+1$ in place of $k$.

Proof.

First we define

[TABLE]

so that the estimate (3.60) immediately follows and the estimate (3.61) follows from Cauchy inequality.

Then, an explicit computation gives

[TABLE]

We remark that, as it can be seen by a qualitative analysis and we will also see quantitatively, the largest contribution to the estimate of the reminder term comes from the term $[U,E^{i}]$ in (3.68), followed (in size, but not in terms of order of magnitude) by the term coming from $R_{\geq m+1}$ still in (3.68). All the other terms admit estimates which of higher order.

Let us prove by induction on $k\geq 0$ estimates (3.59),(3.60) and (3.61)

For $k=0$, one has $N_{\leq 1}\equiv 0$ and $\left\|\underline{\bf R}_{\geq 2}\right\|_{r_{0}}\leq c_{0}r_{0}^{2}=\epsilon_{0}$. Hence, inequalities hold true for $k=0$. Assume that they hold for all $0\leq l\leq k$ and let us prove the inequality for $m=2^{k+1}$.

Since $r_{1}$ and $\delta_{0}$ do not follow the induction definition of $r_{k}$ and $\delta_{k}$, we have to prove separatly the case $k=1$. Since $N_{\leq 1}\equiv 0$ then (3.66) is not present, as well as the last term in (3.68). Furthermore the nonlinear cohomological equation reduces to the linear one, so $U$ can be estimated using Remark 2.10 with $c=1$ which gives

[TABLE]

We have that, by (3.45), (3.40), the families corresponding to (3.65) and (3.67) are estimated by (with a little abuse of notation)

[TABLE]

Concerning (3.68), by (3.46) we have

[TABLE]

and thus

[TABLE]

It follows that

[TABLE]

provided

[TABLE]

From Lemma A.4 it follows that

[TABLE]

We also remark that, by Cauchy estimate, we have

[TABLE]

This concludes the proof of for case $k=1$.

Assume now $k\geq 1$. According to Lemma 3.2, we have $\|\underline{U}(z)\|_{r_{k}}\leq 4\epsilon_{k}$ as soon as

[TABLE]

Hence, by (3.45), (3.40), the above families are estimated by

[TABLE]

Concerning (3.68), by (3.46) we have

[TABLE]

and, by (3.61), (3.60), (3.59) and (3.58), we have

[TABLE]

Summing up we have

[TABLE]

and therefore the same estimate holds for $\left\|\underline{\eqref{r.3}}\right\|_{r_{k}-\delta_{k}}$. Summing up the different contributions, we have

[TABLE]

which, provided

[TABLE]

gives

[TABLE]

ow, from Lemma A.4, since $b\geq 2$, one has

[TABLE]

For $k=1$ (which corresponds to $m=2$), we have

[TABLE]

provided

[TABLE]

which is equivalent to

[TABLE]

For $k\geq 2$ we have

[TABLE]

provided

[TABLE]

which, since the r.h.s. is a decreasing function of $k$, is implied by

[TABLE]

which in turn is implied by (3.76). ∎

From Lemma 3.6, by a completely standard argument, the following Corollary follows

Corollary 2**.**

The sequence of transformations $\{\Psi_{k}\}_{k\geq 1}$ defined by $\Psi_{k}:=\Phi_{2^{k-1}}^{-1}\circ\cdots\circ\Phi_{1}^{-1}$ converges to an analytic transformation $\Psi$ in a neighborhood of the origin and it conjugates the family $\{X^{i}\}_{i\geq 1}$ to a a family of normal forms $\{NF^{i}\}_{i\geq 1}$.

Appendix A A technical Lemma

Lemma A.1**.**

Equation (3.55) holds.

Proof.

Denote by $d_{k}$ the l.h.s. of (3.55), one has

[TABLE]

where we used the formula

[TABLE]

Now, the result immediately follows.∎

Lemma A.2**.**

Equation (3.56) holds.

Proof.

We use the discrete analogue of the formula of the Duhamel formula, namely we make the substitution $r_{k}=d_{k}s_{k}$, where $d_{k}$ was defined in the proof of Lemma A.1. One gets

[TABLE]

and thus

[TABLE]

from which

[TABLE]

Now, one has

[TABLE]

Thus,

[TABLE]

∎

Lemma A.3**.**

Equation (3.58) holds.

Proof.

The first inequality is trivial. We discuss the second one. Using the definition of $r_{k+1}$, we have

[TABLE]

now, one has

[TABLE]

which is of the form $1-e^{-x}$ with $x$ varying from [math] to $\frac{b}{2}\ln 2$. Remarking that in an interval $[0,x_{0}]$ one has

[TABLE]

we get

[TABLE]

and thus, for $k\geq 1$,

[TABLE]

Now one has

[TABLE]

which, for $x=1/2$, gives

[TABLE]

and thus

[TABLE]

adding the first term, namely $\frac{8}{7}\frac{\epsilon_{0}}{r_{0}}$, one gets the thesis immediately follows. ∎

Lemma A.4**.**

Let ${\bf F}$ be a summable normally analytic vector fiels with $F^{i}$ having a zero of order $m$ at the origin for all $i$. Let $0<\alpha\leq 1$, then

[TABLE]

Proof.

Consider the function $\underline{{\bf F}}(z)=\sum_{Q,i}F_{Q,i}z^{Q}\vec{e}_{i}$; since all the coefficients are positive one has, for any $i$,

[TABLE]

Thus one gets

[TABLE]

∎