Invariant curves of smooth quasi-periodic mappings

TL;DR

This paper investigates the existence of invariant curves in smooth quasi-periodic planar mappings, focusing on mappings with multiple frequencies and specific smoothness conditions.

Contribution

It establishes conditions for the existence of invariant curves in quasi-periodic mappings with high smoothness and multiple frequencies, extending previous results in dynamical systems.

Findings

Proves existence of invariant curves under certain smoothness and intersection conditions.

Extends invariant curve theory to mappings with multiple frequencies.

Provides mathematical criteria for invariant curve existence in quasi-periodic systems.

Abstract

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.

Equations

$$\mathfrak{M}:\quad\begin{array}[]{ll}\left\{\begin{array}[]{ll}\theta_{1}=\theta+r+f(\theta,r),\\[5.69046pt] r_{1}=r+g(\theta,r),\\[2.84544pt] \end{array}\right.\ (\theta,r)\in\mathbb{R}\times[a,b],\end{array}$$

$$\mathfrak{M}_{0}:\quad\begin{array}[]{ll}\left\{\begin{array}[]{ll}x_{1}=x+\alpha(y)+\varphi_{1}(x,y),\\[5.69046pt] y_{1}=y+\varphi_{2}(x,y),\end{array}\right.\end{array}$$

$$\mathfrak{M}_{1}:\quad\begin{array}[]{ll}\left\{\begin{array}[]{ll}\theta_{1}=\theta+\beta+r+f(\theta,r),\\[5.69046pt] r_{1}=r+g(\theta,r),\end{array}\right.\ \ \ \ (\theta,r)\in\mathbb{R}\times[a,b],\end{array}$$

$$\Big{|}{\langle k,\omega\rangle{\beta\over{2\pi}}-j}\Big{|}\geq{\gamma\over{|k|^{\tau}}},\ \ \ \ \forall\ \ k\in\mathbb{Z}^{n}\backslash\{0\},\ \ \forall j\in\mathbb{Z},$$

$$\begin{array}[]{ll}\mathcal{M}_{\delta}:\ \ \left\{\begin{array}[]{ll}\theta_{1}=\theta+\beta+\delta l(\theta,r)+\delta f(\theta,r,\delta),\\[5.69046pt] r_{1}=r+\delta m(\theta,r)+\delta g(\theta,r,\delta),\end{array}\right.\ \ \ (\theta,r)\in\mathbb{R}\times[a,b],\end{array}$$

$$\begin{array}[]{ll}x^{\prime\prime}$+$a$$

x^{+}$$-b$$x^{-}$=$f(t),\end{array}$$

$$f(t)=F({\omega }_{1}t,{\omega }_{2}t,\cdots ,{\omega }_{n}t).$$

$$F(\theta )=\sum _{k\in {\mathbb{Z}}^n}{f}_k{e}^{i⟨k,\theta ⟩},$$

$$f(t)=\sum _{k\in {\mathbb{Z}}^n}{f}_k{e}^{i⟨k,\omega ⟩t}.$$

$$\big{|}F\big{|}_{r}=\sup\limits_{\theta\in\Pi_{r}^{n}}|F(\theta)|=\sup\limits_{\theta\in\Pi_{r}^{n}}\Big{|}\sum_{k\in\mathbb{Z}^{n}}f_{k}e^{i\langle k,\theta\rangle}\Big{|}<+\infty.$$

$$\begin{array}[]{ll}{{|\langle k,\omega\rangle|}\geq{c\over{|k|^{\sigma_{0}}}}},\ \ \ \ c,\sigma_{0}>0\end{array}$$

$$\mathfrak{M}(\mathbf{\Gamma })\cap \mathbf{\Gamma }\ne \varnothing$$

$$\underset{T\to +\infty }{\lim }\frac{1}{2T}{\int }_{-T}^{T}rd\theta =\underset{T\to +\infty }{\lim }\frac{1}{2T}{\int }_{-T}^T{r}_{1}d{\theta }_1.$$

$$r=-\frac{\partial }{\partial \theta }H({\theta }_1-\theta ,\theta ),r_1=\frac{\partial }{\partial {\theta }_1}H({\theta }_1-\theta ,\theta ),$$

$$\Delta (t,T)={\int }_t^T{r}_{1}d{\theta }_1-{\int }_t^{T}rd\theta =H({\theta }_1(T)-T,T)-H({\theta }_1(t)-t,t).$$

$$|x|=\max (|\theta |,|r|)\text{mbox}forx=(\theta ,r)\in {\mathbb{R}}^2,$$

$$\displaystyle\big{|}h\big{|}_{\mathbb{R}^{2}}=\sup\limits_{x\in{\mathbb{R}^{2}}}|h(x)|,$$

$$\Vert h{\Vert }_p=\sum _{|k|\le p}\underset{x\in {\mathbb{R}}^2}{\sup }|D^{k}h(x)|$$

$$\Vert h{\Vert }_p=\underset{\begin{array}{c}x\ne y\\ |k|=l\end{array}}{\sup }\frac{|D^{k}h(x)-D^{k}h(y)|}{|x-y{|}^s}+\sum _{|k|\le l}\underset{x\in {\mathbb{R}}^2}{\sup }|D^{k}h(x)|$$

$$\displaystyle D^{k}={{\Big{(}{\partial\over{\partial\theta}}\Big{)}}^{k_{1}}}\circ{{\Big{(}{\partial\over{\partial r}}\Big{)}}^{k_{2}}},\ \ \ |k|=|k_{1}|+|k_{2}|,\ \ \ k=(k_{1},k_{2}).$$

$$\begin{array}[]{ll}\left\{\begin{array}[]{ll}a+12^{-3}\gamma\leq\alpha\leq b-12^{-3}\gamma,\\[11.38092pt] \Big{|}{\langle k,\omega\rangle{\alpha\over{2\pi}}-j}\Big{|}\geq{\gamma\over{|k|^{\tau}}},\ \ \ \ \mbox{for all}\ \ k\in\mathbb{Z}^{n}\backslash\{0\},\ \ \ \ j\in\mathbb{Z}\end{array}\right.\end{array}$$

$$\begin{array}[]{ll}0<\gamma<{1\over 2}\min\{1,\ 12^{3}(b-a)\},\ \ \ \ \tau>n.\end{array}$$

$$\begin{array}[]{ll}|f|_{\mathbb{R}^{2}}+|g|_{\mathbb{R}^{2}}\leq{6^{-(n+1)}\over 3}\ {q\over{300c_{0}}}{{\Big{(}{1\over 72}\Big{)}}^{\tau}}{{\Big{(}{\gamma\over\Gamma(\tau+1)}\Big{)}}^{2}},\end{array}$$

$$\begin{array}[]{ll}\big{\|}f\big{\|}_{p}+\big{\|}g\big{\|}_{p}\leq{6^{-(n+1)}\over 3}\ {{q(1-q)}\over{3600(3c_{1}+c_{2})}}{{\Big{(}{1\over 288}\Big{)}^{\tau}}}{{\Big{(}{\gamma\over\Gamma(\tau+1)}\Big{)}}^{2}},\end{array}$$

$$\begin{array}[]{ll}0<q\leq\min\Big{\{}{{p-2\tau-1}\over{p+1}}\log 2,\ 10^{-2}4^{-\tau}\Big{\}}.\end{array}$$

$$\begin{array}[]{ll}\left\{\begin{array}[]{ll}\theta=\theta^{\prime}+\varphi(\theta^{\prime}),\\[5.69046pt] r=\psi(\theta^{\prime}),\end{array}\right.\end{array}$$

$$\mathfrak{M}{|}_{{\Gamma }_0}:{\theta }_1^{\prime }={\theta }^{\prime }+\alpha .$$

$$\mathfrak{M}{|}_{{\Gamma }_0}:{\theta }_1^{\prime }={\theta }^{\prime }+\alpha$$

$$\mathfrak{M}{|}_{{\Gamma }_0}:{\theta }_1^{\prime }={\theta }^{\prime }+\alpha .$$

$$\begin{array}[]{ll}\left\{\begin{array}[]{ll}\big{|}h_{\delta}\big{|}_{E_{\delta}}\leq c_{0}\big{|}h\big{|}_{{\mathbb{R}}^{2}},\\[11.38092pt] {\big{|}h-h_{\delta}\big{|}_{{\mathbb{R}}^{2}}}\leq c_{1}{\big{\|}h\big{\|}_{p}}{{\delta}^{p}},\\[11.38092pt] {\big{|}h_{\delta}-h_{\delta^{{}^{\prime}}}\big{|}_{E_{\delta}}}\leq c_{2}{\big{\|}h\big{\|}_{p}}{{\delta^{{}^{\prime}}}^{p}}\end{array}\right.\end{array}$$

Full text

au1]hp@mail.bnu.edu.cn

au1]xli@bnu.edu.cn

au3]bliu@pku.edu.cn

1 Introduction

In this paper we are concerned with the existence of invariant curves of the following planar quasi-periodic mappings

[TABLE]

where the perturbations $f(\theta,r)$ and $g(\theta,r)$ are quasi-periodic in $\theta$ with the frequency $\omega=(\omega_{1},\omega_{2}$, $\cdots,\omega_{n})$, $\mathcal{C}^{p}$ smooth in $\theta$ and $r$.

In 1962, Moser [7] considered the twist mapping

[TABLE]

where the perturbations $\varphi_{1},\varphi_{2}$ are assumed to be small and of periodic $2\pi$ in $x.$ He obtained the existence of invariant closed curves of $\mathfrak{M}_{0}$ which is of class $\mathcal{C}^{333}$. About $\mathfrak{M}_{0}$, an analytic version of the invariant curve theorem was presented in [13], a version in class $\mathcal{C}^{5}$ in Rüssmann [10] and a optimal version in class $\mathcal{C}^{p}$ with $p>3$ in Herman [2, 3].

When the perturbations $f(\theta,r),g(\theta,r)$ in (1.1) are quasi-periodic in $\theta$, there are some results about the existence of invariant curves of the following planar quasi-periodic mappings

[TABLE]

where the functions $f(\theta,r)$ and $g(\theta,r)$ are quasi-periodic in $\theta$ with the frequency $\omega=(\omega_{1},\omega_{2}$,$\cdots,\omega_{n})$, real analytic in $\theta$ and $r$, and $\beta$ is a constant.

When the map $\mathfrak{M}_{1}$ in (1.2) is an exact symplectic map, $\omega_{1},\omega_{2},\cdots,\omega_{n}$, $2\pi\beta^{-1}$ are sufficiently incommensurable, Zharnitsky [14] proved the existence of invariant curves of the map $\mathfrak{M}_{1}$ and applied this result to present the boundedness of all solutions of Fermi-Ulam problem. His proof is based on the Lagrangian approach introduced by Moser [9] and used by Levi and Moser in [5] to show a proof of the twist theorem.

When the map $\mathfrak{M}_{1}$ in (1.2) is reversible with respect to the involution $\mathcal{G}:(x,y)\mapsto(-x,y)$, that is, $\mathcal{G}\mathfrak{M}_{1}\mathcal{G}=\mathfrak{M}_{1}^{-1},\omega_{1},\omega_{2},\cdots,\omega_{n},2\pi\beta^{-1}$ satisfy the Diophantine condition

[TABLE]

Liu [6] stated some variants of the invariant curve theorem for quasi-periodic reversible mapping $\mathfrak{M}_{1}$.

In this paper, motivated by the above references, especially by Rüssmann [12], instead of the exact symplecticity or reversibility assumption on $\mathfrak{M}$, we assume that this mapping satisfies the intersection property, and obtain the invariant curve theorem for the quasi-periodic mapping $\mathfrak{M}$ in the smooth case, other than analytic case.

Incidently, in [4] we use this theorem to establish the existence of invariant curves of the planar quasi-periodic mapping

[TABLE]

where the functions $l,m,f,g$ are quasi-periodic in $\theta$ with the frequency $\omega$=$(\omega_{1}$,$\omega_{2}$,

$\cdots$, $\omega_{n}),$ $f(\theta,r,0)=g(\theta,r,0)=0,$ $\beta$ is a constant, $0<\delta<1$ is a small parameter. As an application, we also use them to study the existence of quasi-periodic solutions and the boundedness of all solutions for an asymmetric oscillation

[TABLE]

where $a$, $b$ are two different positive constants, $x^{+}=\max\{x,0\}$, $x^{-}=\max\{-x,0\}$, $f(t)$ is a smooth quasi-periodic function with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$.

Finally, we must point out that in order to obtain the existence of invariant curves for the quasi-periodic mapping $\mathfrak{M}$, we need to assume that this mapping belongs to $\mathcal{C}^{p}$ with $p>2n+1$ and $n$ is the number of the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$. Meanwhile we note that when $n=1$, quasi-periodic mappings are periodic mappings, and the optimal smoothness assumption is $\mathcal{C}^{p}$ with $p>3$. Hence our smoothness assumption for quasi-periodic mappings agrees with that for periodic mappings, and is optimal in this sense.

Our efforts in this paper are same as Rüssmann [12], we are more interested in weak conditions for the perturbations $f,g$ than in high differentiability properties of the constructed invariant curves, and the main line of the proofs is also similar to that of Rüssmann [12].

The rest of the paper is organized as follows. In Section 2, we list some properties of quasi-periodic functions, and then state the main invariant curve theorem (Theorem 2.8) for the quasi-periodic mapping $\mathfrak{M}$ which is given by (1.1). The proofs of Theorem 2.8 are given in Sections 3, 4, 5. In this section 6, we formulate the detail proofs of the Lemma 2.11 which have been used in the previous sections.

2 Quasi-periodic functions and the main result

2.1 The space of quasi-periodic functions

We first define quasi-periodic functions with the frequency $\omega$, here the $n$-dimensional frequency vector $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ is rationally independent, that is, for all $k=(k_{1},k_{2},\cdots,k_{n})\neq 0$, $\langle k,\omega\rangle=\displaystyle\sum\limits_{j=1}^{n}k_{j}\omega_{j}\neq 0$.

Definition 2.1

$f(t)$* is called a continuous quasi-periodic function with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, if there is a continuous function $F(\theta_{1},\theta_{2},\cdots,\theta_{n})$ which is $2\pi$-periodic in each $\theta_{j}\,(1\leq j\leq n)$ such that*

[TABLE]

Moreover, $f(t)$ is called a $\mathcal{C}^{p}$/real analytic quasi-periodic function, if $F$ is $\mathcal{C}^{p}$/real analytic, meanwhile we say that $F$ is a shell function of $f$.

Denote by $Q(\omega)$ the space of real analytic quasi-periodic functions with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$. Given $f(t)\in Q(\omega)$, suppose that the corresponding shell function $F$ has the following Fourier expansion

[TABLE]

which is $2\pi$-periodic in each variable, real analytic and bounded in a complex neighborhood $\Pi_{r}^{n}=\{(\theta_{1},\theta_{2},\cdots,\theta_{n})\in\mathbb{C}^{n}:|\mathrm{Im}\ \theta_{j}|\leq r,j=1,2,\cdots,n\}$ of $\mathbb{R}^{n}$ for some $r>0$. The function $f(t)$ is obtained from $F(\theta)$ by replacing $\theta$ by $\omega t$, and has the following expansion

[TABLE]

Definition 2.2

For $r>0$, let $Q_{r}(\omega)\subseteq Q(\omega)$ be the set of real analytic quasi-periodic functions $f$ such that the corresponding shell functions $F$ are bounded on the subset $\Pi_{r}^{n}$ with the supremum norm

[TABLE]

Also we define the norm of $f$ as \big{|}f\big{|}_{r}=\big{|}F\big{|}_{r}.

The following properties of quasi-periodic functions can be found in [13, chapter 3].

Lemma 2.3

*The following statements are true:

$(i)$ Let $f(t),g(t)\in Q(\omega)$, then $g(t+f(t))\in Q(\omega);$

$(ii)$ Suppose that*

[TABLE]

*for all integer vectors $k\neq 0$. Let $h(t)\in Q(\omega)$ and $\tau=\beta t+h(t)\ (\beta+h^{{}^{\prime}}>0)$,

then the inverse relation is given by $t={\beta^{-1}}\tau+h_{1}(\tau)$ and $h_{1}\in Q({\omega\over\beta})$. In particular, if $\beta=1$, then $h_{1}\in Q(\omega).$*

Throughout this paper, we assume that the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ satisfies the Diophantine condition

[TABLE]

for all integer vectors $k\neq 0$. It is not difficult to show that for $\sigma_{0}>n$, the Lebesgue measure of the set of $\omega$ satisfying the above inequalities is positive for a suitably small $c$.

2.2 The main result

First we give the following definitions.

Definition 2.4

Let $\mathfrak{M}$ be a mapping given by (1.1). It is said that $\mathfrak{M}$ has the intersection property if

[TABLE]

for every curve $\mathbf{\Gamma}:\theta=\xi+\varphi(\xi),\ r=\psi(\xi)$, where the continuous functions $\varphi$ and $\psi$ are quasi-periodic in $\xi$ with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$.

Definition 2.5

Let $\mathfrak{M}$ be a mapping given by (1.1). We say that $\mathfrak{M}:\mathbb{R}\times[a,b]\to\mathbb{R}^{2}$ is an exact symplectic if $\mathfrak{M}$ is symplectic with respect to the usual symplectic structure $dr\wedge d\theta$ and for every curve $\mathbf{\Gamma}:\theta=\xi+\varphi(\xi),\ r=\psi(\xi)$, where the continuous functions $\varphi$ and $\psi$ are quasi-periodic in $\xi$ with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, we have

[TABLE]

We claim that if the mapping $\mathfrak{M}$ is an exact symplectic map, then it has intersection property. In order to prove this result, we first give an useful lemma, and its proof is simple.

Lemma 2.6

If $r=r(\theta)$ is quasi-periodic in $\theta$ and $F(\theta,r)$ is quasi-periodic in $\theta$ with the same frequency, then $F(\theta,r(\theta))$ is also quasi-periodic in $\theta$ with the same frequency.

Now we are going to prove the following lemma.

Lemma 2.7

If the mapping $\mathfrak{M}$ is an exact symplectic map, then it has intersection property.

Proof. Since the mapping $\mathfrak{M}$ is exact symplectic and it is also a monotonic twist map, according to the paper by Zharnitsky [14], there is a function $H$ such that the mapping $\mathfrak{M}$ can be written by

[TABLE]

where $H$ is quasi-periodic in the second variable.

Now we prove the intersection property of the mapping $\mathfrak{M}$, that is, given any continuous quasi-periodic curve $\Gamma:r=r(\theta)$, we need to prove that $\mathfrak{M}(\Gamma)\cap\Gamma\neq\emptyset.$ Define two sets $\mathbb{B}$ and ${\mathbb{B}}_{1}$ : the set $\mathbb{B}$ is bounded by four curves \big{\{}(\theta,r):\theta=t\big{\}}, \big{\{}(\theta,r):\theta=T\big{\}}, \big{\{}(\theta,r):r=r_{*}\big{\}} and \big{\{}(\theta,r):r=r(\theta)\big{\}}, the set ${\mathbb{B}}_{1}$ is bounded by four curves \big{\{}(\theta,r):\theta=t\big{\}}, \big{\{}(\theta,r):\theta=T\big{\}}, \big{\{}(\theta,r):r=r_{*}\big{\}} and the image of $\Gamma$ under $\mathfrak{M}$. Here we choose $r_{*}<\min r(\theta)$. It is easy to show that the difference of the areas of ${\mathbb{B}}_{1}$ and $\mathbb{B}$ is

[TABLE]

From the definition of $\mathfrak{M}$ and Lemma 2.6, we know that $\theta_{1}(T)-T=r(T)+f(T,r(T))$ is quasi-periodic in $T$ and $\theta_{1}(t)-t=r(t)+f(t,r(t))$ is quasi-periodic in $t$. Hence using Lemma 2.6 again, it follows that $\Delta(t,T)$ is quasi-periodic in $t$ and $T$.

Hence there are at least two pairs of $(t_{1},T_{1})$ and $(t_{2},T_{2})$ such that $\Delta(t_{1},T_{1})<0,\Delta(t_{2},T_{2})>0.$ The intersection property of $\mathfrak{M}$ follows from this fact, which proves the lemma. $\square$

For the quasi-periodic mapping $\mathfrak{M}$ we assume that $f,g:\mathbb{R}^{2}\mapsto\mathbb{R}$ are of class $\mathcal{C}^{p}$, and define

[TABLE]

if $p\geq 0$ is an integer, and

[TABLE]

if $p=l+s$,  $l\geq 0$ is an integer, $s\in(0,1)$, where

[TABLE]

We choose a rotation number $\alpha$ satisfying the inequalities

[TABLE]

with some constants $\gamma,\tau$ satisfying

[TABLE]

Now we are in a position to state our main result.

Theorem 2.8

Suppose that the quasi-periodic mapping $\mathfrak{M}$ given by (1.1) is of class $\mathcal{C}^{p}\ (p>2\tau+1)$, and satisfies the intersection property, the functions $f(\theta,r),g(\theta,r)$ are quasi-periodic in $\theta$ with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, and satisfy the following smallness conditions

[TABLE]

[TABLE]

where $\Gamma$ is the Gamma function, $\gamma,\tau$ satisfy (2.3), $c_{0},c_{1},c_{2}$ are positive constants depending only on $p$ and $\omega$, and $q$ is a number satisfying

[TABLE]

Then for any number $\alpha$ satisfying the inequalities (2.2), the quasi-periodic mapping $\mathfrak{M}$ has an invariant curve $\Gamma_{0}$ with the form

[TABLE]

where $\varphi,\psi$ are quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, and the invariant curve $\Gamma_{0}$ is continuous and quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$. Moreover, the restriction of $\mathfrak{M}$ onto $\Gamma_{0}$ is

[TABLE]

Remark 2.9

Here we assume that the mapping $\mathfrak{M}$ is of class $\mathcal{C}^{p}$ with $p>2\tau+1>2n+1$. $n=1$ corresponds to the periodic case, in which $p>3$ is the optimal smoothness condition. Hence our smoothness assumption for quasi-periodic mappings is optimal in this sense.

Remark 2.10

If all conditions of Theorem 2.8 hold, then the mapping $\mathfrak{M}$ has many invariant curves ${\Gamma_{0}}$, which can be labeled by the form

[TABLE]

of the restriction of $\mathfrak{M}$ onto ${\Gamma_{0}}.$ In fact, given any $\alpha$ satisfying the inequalities (2.2), there exists an invariant curve ${\Gamma_{0}}$ of $\mathfrak{M}$ which is quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, and the restriction of $\mathfrak{M}$ onto ${\Gamma_{0}}$ has the form

[TABLE]

The existence of such $\alpha$ can be found in Lemma 2.12.

The constants $c_{0},c_{1},c_{2}$ in the main result depend on how well functions of class $\mathcal{C}^{p}$ can be approximated by analytic ones.

Lemma 2.11

Let $h(\cdot,y)\in\mathcal{C}^{p}$ be a quasi-periodic function with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, then for any $\delta>0$, there exists a holomorphic function $h_{\delta}:{\mathbb{C}}^{2}\mapsto\mathbb{C},$ $h_{\delta}(\cdot,y)\in Q_{r}(\omega)$, $h_{\delta}({\mathbb{R}}^{2})\subseteq\mathbb{R}$ such that the following inequalities

[TABLE]

hold for $0<\delta<{\delta^{{}^{\prime}}}$, where

[TABLE]

$c_{0},c_{1},c_{2}$* are positive constants only depending on $p,\omega$.*

The detail proof of Lemma 2.11 is given in the Appendix. The proof of Lemma 2.11 is similar to the periodic case. When $h\in\mathcal{C}^{p}$ is a periodic function, there are some detail proofs of Lemma 2.11 available in the literature, for example, see Moser [8, p. 528-529],  Rüssmann [10, p. 74-78],  Zehnder [15, p. 110-113].

2.3 The measure estimate

Lemma 2.12

If $\tau>n$, then for suitable small $\gamma$, the set of $\alpha$ satisfying (2.2) has positive measure.

Proof: Choose some $n$-dimensional frequency vector $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ satisfying (2.1) and let $\mathcal{D}_{\gamma,\tau}^{\omega}$ denote the set of all $\alpha\in\mathbb{R}$ satisfying (2.2) with the fixed $\gamma$ and $\tau$. Then $\mathcal{D}_{\gamma,\tau}^{\omega}$ is the complement of the open dense set $\mathcal{R}_{\gamma,\tau}^{\omega}$, where

[TABLE]

Now we estimate the measure of the set $\mathcal{R}_{\omega,\gamma,\tau}^{k,j}$. Set $|k_{\max}|=\max\limits_{1\leq i\leq n}|k_{i}|$, then there exists some $1\leq m\leq n$ such that $|k_{m}|=|k_{\max}|$, and $1\leq{{|k|}\over{|k_{\max}|}}\leq n.$ Therefore, we have

[TABLE]

where b_{j}={1\over{k_{\max}|\omega_{m}|}}\Big{\{}\sum\limits_{i\not=m}k_{i}\omega_{i}\alpha-2\pi j\Big{\}} and $\delta_{k}={2\pi\gamma\over{|k|^{\tau}}}\,{1\over{|k_{\max}||\omega_{m}|}}.$ Hence,

[TABLE]

Since $1\leq{{|k|}\over{k_{\max}}}\leq n,$ then we have the following measure estimate

[TABLE]

Next we estimate the measure of the set $\mathcal{R}_{\gamma,\tau}^{\omega}$. Since for $\alpha\in\mathcal{R}_{\omega,\gamma,\tau}^{k,j}$,

[TABLE]

then we have

[TABLE]

where $c_{0}$ is a constant independent of $k$. Thus

[TABLE]

Also, if $\tau>n$,

[TABLE]

Hence, for any $\tau>n$,

[TABLE]

and

[TABLE]

This completes the proof. $\square$

3 The iteration process

In this section we present an iteration process leading to the proof of Theorem 2.8.

Firstly, we introduce new variables by the linear transformation

[TABLE]

where $\alpha$ is the chosen rotation number satisfying (2.2), ${\varepsilon_{0}}$ is defined by

[TABLE]

In the new coordinates the given mapping (1.1) having the intersection property in the strip $S=\{(\theta,r)\in{\mathbb{R}}^{2}:a<r<b\}$ gets the form

[TABLE]

Clearly the intersection property is preserved and holds in the strip

[TABLE]

where we have used (3.1) and ${\Gamma(\tau+1)}\geq 1$ for $\tau>n$.

Since the $\mathcal{C}^{p}$ functions $f(\cdot,\alpha+{\varepsilon_{0}}y),g(\cdot,\alpha+{\varepsilon_{0}}y)$ are quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, by assumption we may apply Lemma 2.11 to obtain a family of holomorphic functions $f_{\delta}(\cdot,\alpha+{\varepsilon_{0}}y),\ g_{\delta}(\cdot,\alpha+{\varepsilon_{0}}y)\in Q_{r}(\omega)\ (\delta>0)$, with which we define the quasi-periodic mappings

[TABLE]

Define

[TABLE]

and a sequence

[TABLE]

where $q$ is a real number satisfying (2.6), and set

[TABLE]

Then the estimates of Lemma 2.11 can be written in the form

[TABLE]

where the mapping

[TABLE]

Before we describe the iteration process, some definitions and notations are useful.

(i) Given subsets $D_{1},\cdots,D_{\ell}$ of ${\mathbb{C}^{m}}$ and functions $F_{j}:D_{j}\mapsto{\mathbb{C}^{m}},\ j=1,\cdots,\ell-1$. Then

[TABLE]

exists if

[TABLE]

In the case $F_{j}=\mbox{id}=\mbox{id}_{D_{j}}$ this condition means

[TABLE]

(ii) For $r,\ s>0$, define

[TABLE]

(iii) For $D=D(r,s)$, denote by $T(D)=T(r,s)$ the set of all holomorphic functions $F:D\mapsto{\mathbb{C}^{2}}$ satisfying the identity

[TABLE]

where $\sigma$ is defined by

[TABLE]

for all $(x,y)\in{\mathbb{C}^{2}}$ with $x=a+bi$, $\bar{x}=a-bi,a,b\in\mathbb{R}$.

(iv) Define the mappings $\Omega_{k}\ (k=0,1,\cdots)$ by

[TABLE]

(v) In ${\mathbb{C}^{m}}\ (m=1,2,\cdots)$, define the norm

[TABLE]

(vi) Given $f:D\mapsto{\mathbb{C}^{m}}$ with $D=D(r,s)$,  define

[TABLE]

Now we are going back to the quasi-periodic mappings $A_{k}:E_{k}\mapsto{\mathbb{C}^{2}}$ defined above. We try to fix domains

[TABLE]

and to find mappings $Z_{k}\in T(D^{\prime}_{k}),H_{k}\in T(D_{k})$, and $Z_{k}-\mbox{id},H_{k}-\Omega_{k}$ are quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ in the first variable, such that the diagrams

[TABLE]

exist and commute for $k=0,1,\cdots.$

A proper choice for the constants $r_{k},\ s_{k},\ r^{\prime}_{k},\ s^{\prime}_{k}$ is

[TABLE]

Then obviously $D_{k}\subseteq D^{\prime}_{k}\ (k=0,1,\cdots).$ About the mappings $Z_{k},\ H_{k}$ the following relations are needed

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $b_{k}=2^{-k\tau}(1-q)^{k}$, $B_{k}=(1+q)^{k}$, $M_{k}=2^{-k(\tau+1)}M_{0},\ M_{0}={1\over 3}q\varepsilon_{0}s_{0}.$

Finally define $Z_{0}=\mbox{id}_{D^{\prime}_{0}}\in T(D^{\prime}_{0})$, then by means of this iteration process, if it exists, the assertion of Theorem 2.8 can easily be proved.

In fact, from (3.6), $(3.8)_{k}$ and $(3.9)_{k}$, the sequence $Z_{0},Z_{1},\cdots$ converges uniformly on $\mathbb{R}\times\{0\}$ and the limit $Z_{\infty}(\xi)=\lim\limits_{k\rightarrow\infty}Z_{k}(\xi,0)$ is continuous on $\mathbb{R}$.

Since $Z_{k}\in T(D^{\prime}_{k})\ (k=0,1,\cdots)$,

[TABLE]

Now the commutativity of $(3.5)_{k}$ yields

[TABLE]

Hence

[TABLE]

and by virtue of $(3.8)_{k},(3.10)_{k}$ consequently

[TABLE]

Passing to the limit we get

[TABLE]

in view of (3.3), (3.4) and $B_{k}M_{k}\rightarrow 0.$

Therefore, we can obtain the existence of invariant curves of the mapping $A$, and from the relationship between the mappings $A$ and $\mathfrak{M}$, one can also get the existence of invariant curves of the mapping $\mathfrak{M}$.

From the above analysis, firstly we need to prove the assertion

[TABLE]

and the estimate $(3.9)_{k}$ for $k=0,1,\cdots$.

The proofs of $(3.9)_{k}$ and $(3.11)_{k}\ (k=0,1,\cdots)$ are done by the complete induction. Let us first consider the case $(3.11)_{0}.$ As a consequence of the definition of $Z_{0}$, the relations $(3.7)_{0}$ and $(3.8)_{0}$ are obvious. Moreover if we define H_{0}=A_{0}\big{|}_{D_{0}}, by virtue of $(\ref{b5}),\ (\ref{c2}),\ (\ref{c7}),\ (3.6)$, then

[TABLE]

which is the wanted estimate $(3.10)_{0}$.

From the definition of $\varepsilon_{0}$ and $q$, we have

[TABLE]

Hence

[TABLE]

From the definitions of $D_{0}$ and $E_{0}$, $D_{0}\subseteq E_{0}$ holds. Therefore

[TABLE]

Thus, the diagram $(3.5)_{0}$ exists and commutes.

Now let us suppose that $(3.11)_{k}$ is true for some $k\geq 0.$ We have to show $(3.11)_{k+1}$ and $(3.9)_{k}.$ On this way the crucial result is the construction of the commuting diagram

[TABLE]

with

[TABLE]

and mappings $W_{k}\in T(D^{\prime}_{k+1}),\ \Phi_{k+1}\in T(D_{k+1})$, and $W_{k}-\mbox{id},\Phi_{k+1}-\Omega_{k+1}$ are quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ in the first variable.

The existence of the commuting diagram (3.12) is guaranteed by the inductive theorem (Theorem 5.3), which we will prove in Section 5. This theorem also gives the following estimates

[TABLE]

[TABLE]

[TABLE]

where $Q_{k}$ is a polynomial of degree 2 in the second variable only

[TABLE]

With these assertions of the inductive theorem we can show $(3.11)_{k+1}$ and $(3.9)_{k}.$ From the diagrams $(3.5)_{k}$ and (3.12) we see that

[TABLE]

Define

[TABLE]

of course, $Z_{k+1}\in T(D^{\prime}_{k+1})$, $Z_{k+1}-\mbox{id}$ is quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ in the first variable, and $(3.7)_{k+1}$ holds. The proof of $(3.8)_{k+1}$ by means of $(3.8)_{k}$ and (3.13) is obvious if we notice $W_{k}(D^{\prime}_{k+1})\subseteq D^{\prime}_{k}$ in (3.12). The inequality $(3.9)_{k}$ follows from $(3.8)_{k}$, $(3.14)$ and $W_{k}(D_{k+1})\subseteq D^{\prime}_{k}$ as a consequence of $D_{k+1}\subseteq D^{\prime}_{k+1}.$ We also need to prove

[TABLE]

In fact,

[TABLE]

where $Z_{k+1}$ is real for real arguments as an element of $T^{\prime}(D_{k+1})$, we get

[TABLE]

hence

[TABLE]

Using $(\ref{c6})$, $(3.6)$, $(3.8)_{k}$ we obtain

[TABLE]

By the definition of $E_{k}\ (k=0,1,2,\cdots)$, $Z_{k+1}(D_{k+1})\subseteq E_{k+1}.$

With these assertions we obtain the following commuting diagram

[TABLE]

Comparing the diagrams $(3.5)_{k+1}$ and (3.17), it remains to prove that we can replace $\Phi_{k+1}$ by $H_{k+1}$ if we replace A_{k}\big{|}_{E_{k+1}} by $A_{k+1}.$ Moreover we have to show $(3.10)_{k+1}$, which is not possible without going back to the original quasi-periodic mapping $A$ in order to use the intersection property, and to estimate the polynomial (3.16) well enough such that $(3.10)_{k+1}$ follows from (3.15).

In the following, we will prove these assertions. First of all we give some useful definitions and lemmas. $\mathbb{F}$ stands for $\mathbb{C}$ or $\mathbb{R}$, we call a function analytic if it is holomorphic in the case $\mathbb{F}=\mathbb{C}$ and $\mathbb{R}$-analytic in the case $\mathbb{F}=\mathbb{R}.$ Moreover, for $D\subseteq\mathbb{F}^{m}$ and $d>0$, define the set

[TABLE]

which may be empty. If $D$ is open so is $D-d.$

Lemma 3.1** (Lemma 2 in [12])**

Let $D$ be an open subset of $\mathbb{F}^{m}$, and $F:D\mapsto F(D)\subseteq\mathbb{F}^{m}$ be an analytic mapping satisfying the estimate

[TABLE]

for all $z,\ z^{\prime}\in D$ and some $b>0.$ Then $F(D)$ is open, and the inverse mapping $F^{-1}:F(D)\mapsto D$ exists and is analytic. Moreover for any $d>0$ we have

[TABLE]

It is useful to introduce an arbitrary bijection $\Lambda:\mathbb{F}^{m}\rightarrow\mathbb{F}^{m}$ and to denote by $\Delta=\Delta(\Lambda,\mathbb{F}^{m})$ the set of all subsets $D$ of $\mathbb{F}^{m}$ which are invariant under $\Lambda.$ Furthermore denote by $\Sigma=\Sigma(\Lambda,\mathbb{F}^{m})$ the class of all functions $F:D\mapsto\mathbb{F}^{m}$ such that D\in\Delta,\ \Lambda\circ F=F\circ\Lambda\big{|}_{D}. Clearly, $F(D)\in\Delta$ for a function of class $\Sigma$ which domain is $D,$ and if $F$ is injective, then also $F^{-1}:F(D)\mapsto\mathbb{F}^{m}$ is of class $\Sigma.$ Moreover if $F:D\mapsto\mathbb{F}^{m},$ $G:E\mapsto\mathbb{F}^{m}$ are of class $\Sigma,$ and if $G(E)\subseteq D$, then $F\circ G$ is of class $\Sigma.$

Lemma 3.2** (Theorem 3 in [12])**

Let $D,\ D^{\prime},\ E$ be open subsets of $\mathbb{F}^{m}$ belonging to $\Delta$ with $D\subseteq D^{\prime}$ and $A^{\prime},\ Z,\ \Phi^{\prime}$ be analytic mappings of class $\Sigma$ such that the diagram

[TABLE]

exists and commutes with some $d>0,$ and the estimate

[TABLE]

holds for all $\zeta,\zeta^{\prime}\in D^{\prime}$ with some $b>0.$ Then for any continuous mapping $A^{\prime\prime}:E\mapsto\mathbb{F}^{m}$ of class $\Sigma$ satisfying the estimate

[TABLE]

there exists a continuous mapping $\Phi^{\prime\prime}$ of a class $\Sigma$ such that the diagram

[TABLE]

exists and commutes, and the estimate

[TABLE]

is valid. If $A^{\prime\prime}$ is analytic so is $\Phi^{\prime\prime}.$

We apply Lemma 3.2 to the diagram (3.17) in order to obtain $(3.5)_{k+1}.$ Set

[TABLE]

[TABLE]

[TABLE]

Moreover, $\Delta$ is the set of all subsets of $\mathbb{C}^{2}$ which are invariant under $\Lambda=\sigma$ such that $\Sigma$ is the class of all functions $F:D\mapsto\mathbb{C}^{2}$ with $D\in\Delta$ and $\sigma\circ F=F\circ\sigma.$ Then $D_{k+1},\ D^{\prime}_{k+1},\ E_{k+1}$ are open sets belonging to $\Delta,$ and (3.21) represents analytic functions of class $\Sigma.$ For $A^{\prime}$ and $A^{\prime\prime}$ this follows from Lemma 2.11, for $\Phi^{\prime}$ and $Z$ this is true because of $\Phi_{k+1}\in T(D_{k+1}),Z_{k+1}\in T(D^{\prime}_{k+1}).$ In addition $(3.8)_{k+1}$ is valid.  Therefore Lemma 3.2 can be applied and gives a function $H_{k+1}=\Phi^{\prime\prime}\in T(D_{k+1})$, $H_{k+1}-\Omega_{k+1}$ is quasi-periodic with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$ in the first variable, such that the diagram $(3.5)_{k+1}$ exists and commutes.

We also need to prove $H_{k+1}$ satisfies $(3.10)_{k+1}$. By $(\ref{b6}),\ (\ref{c2}),\ (\ref{c6}),\ (\ref{c7})$, we obtain

[TABLE]

The necessary condition which we have to require is

[TABLE]

for then we get the estimate

[TABLE]

By the definitions of $s_{0},\ b_{k+1},\ M_{k+1}$ in (3.6), $(3.8)_{k+1}$ and $(3.10)_{k+1},$ we have

[TABLE]

hence

[TABLE]

Since $M_{k+1}s_{k}^{-1}\leq M_{0}s_{0}^{-1}\leq{3^{-1}}$, and put $d={1\over 7}s_{k},$ we get

[TABLE]

According to Lemma 3.2, we have

[TABLE]

By $(3.15)$, we get

[TABLE]

then we have the estimate

[TABLE]

with a polynomial $Q_{k}$ defined in (3.16).

In order to obtain a proper estimate for $Q_{k}$, we apply Lemma 3.2 once more to the diagram (3.17), where this time we restrict $D_{k+1}$ to $D=\mathbb{R}^{2}\cap D_{k+1}$ such that we consider (3.18) with

[TABLE]

Here we put \Lambda=\sigma\big{|}_{\mathbb{R}^{2}} such that $D,\ D^{\prime},\ E$ are open subsets of $\mathbb{R}^{2}$ belonging to $\Delta,$ and $A^{\prime},\ \Phi,\ Z$ are analytic functions of class $\Sigma.$ Also the original quasi-periodic mapping $A$ defined at the beginning of Section 3 is of class $\Sigma,$ and it is continuous. Therefore using $(3.7)_{k+1}$, Lemma 3.2 is again applicable and we obtain a continuous function $\Psi_{k+1}=\Phi^{\prime\prime}$ of class $\Sigma$ such that the diagram

[TABLE]

exists and commutes provided (3.19) can be satisfied. Furthermore by means of (3.4), (3.20) and (3.23) we get the estimate

[TABLE]

which leads with (3.15) to

[TABLE]

We recall that the quasi-periodic mapping $A$ has the intersection property at least in the strip (3.2). We apply this property to the family of curves

[TABLE]

where it is clear that these curves lie in $S^{\star}.$ Moreover these curves satisfy the conditions of Definition 2.4. For each $\eta$ with $-s_{k+1}<\eta<s_{k+1}$, there are real numbers $\xi_{0},\ \xi_{1}$ such that

[TABLE]

On the other hand from the commuting diagram (3.24) we have

[TABLE]

The mapping $Z_{k+1}$ is analytic, and as a consequence of $(3.8)_{k+1}$ it is injective. Thus the injectivity of $Z_{k+1}$ yields

[TABLE]

hence

[TABLE]

where (2) indicates the second component of a vector.

This equation leads to a reasonable estimate for the polynomial (3.16). Since we use the maximum norm we get for $-s_{k+1}<\eta<s_{k+1}$ with the notation

[TABLE]

the estimate

[TABLE]

holds by virtue of (3.25), where we put

[TABLE]

Then we have

[TABLE]

For $\eta=0$, we obtain $|a|\leq N$. Therefore we get

[TABLE]

Let $\eta=\pm\tilde{\sigma},\ 0<\tilde{\sigma}<s_{k+1}$, we get

[TABLE]

Letting $\tilde{\sigma}\rightarrow s_{k+1}$, we have

[TABLE]

for all $\eta\in\mathbb{C},\ |\eta|<s_{k+1}$, hence

[TABLE]

In the previous setting, we have obtained

[TABLE]

hence

[TABLE]

This inequality obviously gives the wanted estimate $(3.10)_{k+1}$.

The proof by induction for justifying the iteration process has finished. It remains to find a better form for condition (3.22). Equivalently we may write

[TABLE]

for $0<q\leq{1\over 4},\ \ p>2\tau+1>2n+1>3$, we have ${d^{2}f\over{d{q^{2}}}}(q)\leq 0,$ hence

[TABLE]

As a consequence

[TABLE]

is sufficient for (3.22). This is one of the conditions for $q$ appearing in (2.6).

4 Linear difference equations

In this section we will solve the difference equations

[TABLE]

which plays a central role in the proof of the inductive theorem. Here the mean value of the function $g(x,y)$ over the variable $x$ is defined by $[g](y)=\lim\limits_{T\rightarrow\infty}{1\over T}\int_{0}^{T}g(x,y)dx$, and $\alpha$ is a real number satisfying the Diophantine inequalities

[TABLE]

The functions $f(\cdot,y),g(\cdot,y)\in Q_{r}(\omega)$ are given holomorphic functions of the complex variables $x,y$, and $u,v$ are wanted holomorphic functions of the complex variables $x,y.$ $\varepsilon$ is a positive constant to be determined in such a way that the functions $u,\ v$ will be of the same size.

In order to get estimates for $u,v$ which are good enough for the proof of Theorem 2.8, some technical preparations have to be made.

Lemma 4.1** (Lemma 3.3 in [11])**

Let $\bar{\omega}=(\bar{\omega}_{1},\bar{\omega}_{2},\cdots,\bar{\omega}_{\ell})\in{\mathbb{R}^{\ell}}$ satisfying the inequalities  $D(k,\bar{\omega})\geq\psi(|k|)$, where  D(k,\bar{\omega})=\min\limits_{{j\in\mathbb{Z}}}\Big{|}\Big{\langle}(k,j),\bar{\omega}\Big{\rangle}\Big{|},k\in{\mathbb{Z}^{\ell-1}\backslash\{0\}}, $\psi$ is an approximation function. Then for  $m=1,2,\cdots,$ we have

[TABLE]

where $\bar{k}=(k,j)$.

If we choose

[TABLE]

then by Lemma 4.1 and the Diophantine inequalities (4.2), we obtain

[TABLE]

Meanwhile,

[TABLE]

Therefore

[TABLE]

Lemma 4.2

For $r>0$, let f:\big{\{}x\in\mathbb{C}:|\mathrm{Im}\ x|<r\big{\}}\mapsto\ \mathbb{C} be a holomorphic function and $f\in Q_{r}(\omega)$. Then we have the estimate

[TABLE]

where

[TABLE]

are the Fourier coefficients of $f$,  $F(\theta)$ is the shell function of $f$ and

[TABLE]

Proof: The Fourier coefficients $f_{k}$ of $f$ are given by

[TABLE]

For every $\lambda\in\mathbb{R}^{n}$ with $|\lambda|=\max\limits_{1\leq j\leq n}|\lambda_{j}|<r,$ the function $x\mapsto f(x+i\lambda)$ which domain is r-|\lambda|,\ its Fourier coefficients are

[TABLE]

By Bessel’s inequality,

[TABLE]

Hence

[TABLE]

Define a new function

[TABLE]

then

[TABLE]

Since $F(\theta+i\lambda)e^{-i\langle k,\theta+i\lambda\rangle}$ is $2\pi$-periodic in $\theta_{j}\ (j=1,2,\cdots,n)$, then

[TABLE]

Hence the function ${f_{k}(\lambda)e^{\langle k,\lambda\rangle}}$ is independent of $\lambda$, and

[TABLE]

and consequently

[TABLE]

Finally, by (4.4), we have

[TABLE]

Define $e_{i}\in\mathbb{R}^{n}\ (i=1,2,\cdots,2^{n})$ which have components $\pm 1$, and

[TABLE]

Then

[TABLE]

Let $\lambda=se_{i}$ in (4.5), we obtain

[TABLE]

Passing to the limit $s\rightarrow r$ yields

[TABLE]

Adding these inequalities and by (4.6), we have

[TABLE]

The proof of this lemma is completed. $\square$

Definition 4.3

(i) For $D=D(r,s)$, denote by $P^{m}(r,s)=P^{m}(D)\ (m=1,2,\cdots)$ the linear space of all holomorphic functions $f:D\rightarrow\mathbb{C}^{m}$ satisfying

[TABLE]

*Clearly $f,g\in T(D)$ implies $f-g\in P^{2}(D).$

(ii) For a function $f\in P^{m}(D)$, denote its mean value over the variable $x$ by*

[TABLE]

Theorem 4.4

Let $\alpha$ be a real number satisfying (4.2), and $f(\cdot,y)\in Q_{r}(\omega)$ be a function belonging to $P^{1}(r,s)$ for some positive constants $r,s.$ Then the difference equation

[TABLE]

has a unique solution $u\in Q(\omega),\ u\in P^{1}(r,s)$ with $[u]=0.$ For this solution the estimate

[TABLE]

holds, where $\varepsilon$ is defined by

[TABLE]

Proof: Since the restriction of $f(x,y)$ onto $\mathbb{R}^{2}$ is a continuously differentiable and quasi-periodic function in $x$, it can be expanded into its Fourier series

[TABLE]

where

[TABLE]

are the Fourier coefficients of $f(x,y)$.  An application of Lemma 4.2 to the restriction of $f(x,y)$ onto $D(r,s)$ yields that

[TABLE]

Let

[TABLE]

After straightforward calculations we obtain the relation between Fourier coefficients $f_{k}(y)$ and $u_{k}(y)$ as follows

[TABLE]

then

[TABLE]

which is the uniquely determined Fourier expansion of the wanted solution $u$ satisfying $u\in Q(\omega)$ with $[u]=0.$

Firstly, we estimate the sum

[TABLE]

By Cauchy-Schwarz inequality, we have

[TABLE]

By $(\ref{d3}),\ (\ref{d13}),$ we obtain

[TABLE]

[TABLE]

hence

[TABLE]

Set $g_{0}(y)=0$, we get

[TABLE]

Letting $N\rightarrow\infty$, we have

[TABLE]

For the last series we get the estimate

[TABLE]

Hence

[TABLE]

which completes the proof of the lemma. $\square$

Now we are ready to solve equation (4.1).

Theorem 4.5

Let $\alpha$ be a real number satisfying (4.2), and $f(\cdot,y),g(\cdot,y)\in Q_{r}(\omega)$ be functions belonging to $P^{1}(r,s)$ and satisfying the estimates

[TABLE]

with some positive constants $r,\ s,\ M.$ Then the difference equations (4.1) with $\varepsilon$ defined in (4.8) have a unique solution $u,v\in Q(\omega),$ $u,v\in P^{1}(r,s)$ with $[u]=0.$ For this solution the estimates

[TABLE]

[TABLE]

are valid for $0<2\rho<r$.

Proof: In the first equation of (4.1) the mean value must vanish on both sides. Hence we get the condition

[TABLE]

for the mean value of $v$. As a consequence, we have $[v]\in P^{1}(r,s)$ and

[TABLE]

in view of (4.10). Theorem 4.4 gives a unique solution $v=\tilde{v}\in P^{1}(r,s)$ of the second equation of (4.1) with $[\tilde{v}]=0.$ This solution has the estimate

[TABLE]

because of (4.10). Define $v=\tilde{v}+[v]$, we obtain the uniquely determined solution $v\in P^{1}(r,s)$ of the second equation of (4.1) satisfying (4.13). This solution has the estimate (4.12) as a consequence of (4.14) and (4.15).

Define $h=\varepsilon\tilde{v}+f$, note that $\tilde{v}$ is defined in $D(r-\rho,s)$, then $h$ is well defined in $D(r-\rho,s)$. As a consequence we have

[TABLE]

and

[TABLE]

Hence, the first equation of (4.1) can be rewritten in the form

[TABLE]

Thus Theorem 4.4 gives a uniquely determined solution $u\in P^{1}(r,s)$ of (4.17) with $[u]=0.$ For an estimate of $u$ we apply Theorem 4.4 to (4.17) restricted to $D(r-\rho,s)$ such that in (4.7) we have to replace $f$ by $h$ and $r$ by $r-\rho.$ Then (4.11) follows by means of (4.16). The proof is finished. $\square$

5 The inductive theorem

First of all we give together constants, domains, and mappings appearing in the formulation of the inductive theorem.

(I) Constants and their relations

We introduce the constants

[TABLE]

and the auxiliary constants $\theta,\rho$ satisfying the relations

[TABLE]

(II) Domains and Mappings

Choose

[TABLE]

[TABLE]

and introduce the mappings

[TABLE]

[TABLE]

[TABLE]

for all $(x,y)\in\mathbb{C}^{2},$ where we use the same symbol for the mappings $\Omega,\ \Omega_{+},\Theta$ as well as for their restrictions to subsets of $\mathbb{C}^{2}.$

For the proof of the inductive theorem we need two useful lemmas.

Lemma 5.1** (Lemma 5 in [12])**

Let $D\subseteq\mathbb{C}^{\ell}$ be an open and convex set, and $F:D\mapsto\mathbb{C}^{m}$ be a holomorphic function. Then for any $d>0$, we have

[TABLE]

Lemma 5.2** (Lemma 6 in [12])**

For some $r>0,$ let f:\big{\{}z\in\mathbb{C}:|z|<r\big{\}}\mapsto\mathbb{C} be a holomorphic function with power series expansion

[TABLE]

Then for the polynomial

[TABLE]

of degree $m-1$ depending on $q\ (0<q<1)$, we have the estimate

[TABLE]

Theorem 5.3** (Inductive Theorem)**

Let constants (5.1) and auxiliary constants $\theta,\rho$ be given such that the relations in (I) are satisfied, and let domains $D,\ D_{+},\ D^{\prime},\ D^{\prime}_{+},\ D_{+}^{\star}$ and mappings $\Omega,\ \Omega_{+},\ \Theta$ be given as in (II). Then for any mapping

[TABLE]

$H-\Omega$* is quasi-periodic with the frequency $\omega$ in the first variable and satisfying*

[TABLE]

there are mappings $W\in T(D^{\prime}_{+})$, $\Phi_{+}\in T(D_{+})$ such that the diagram

[TABLE]

exists and commutes. Moreover $W-\Theta,\Phi_{+}-\Omega_{+}-Q$ are quasi-periodic with the frequency $\omega$ in the first variable and the following estimates are satisfied

[TABLE]

[TABLE]

[TABLE]

where $Q:D_{+}\mapsto\mathbb{C}^{2}$ is defined by

[TABLE]

with some constants $a_{0},\ a_{1},\ a_{2}\in\mathbb{R}.$

Proof:  1) First of all, define

[TABLE]

and by assumption, $H-\Omega$ is quasi-periodic with the frequency $\omega$ in the first variable and $h\in P^{2}(D).$

The results of Section 4 enable us to solve the linear difference equations

[TABLE]

Let $d\,\Omega$ be the differential of $\Omega$ and define

[TABLE]

[TABLE]

[TABLE]

With these definitions and notations, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence, the difference equations (5.9) can be written in the more compact form

[TABLE]

Define

[TABLE]

we ought to show $w\in P^{2}(D),\ \phi\in P^{2}(D_{+}).$

After having obtained a solution $w\in P^{2}(D^{\prime}_{+})$ of (5.10), we try to determine $\phi$ from the equation

[TABLE]

which holds because the diagram in the inductive theorem can commute.

Now (5.11) can be rewritten in the form

[TABLE]

which is

[TABLE]

First,

[TABLE]

[TABLE]

Since ${{\varepsilon_{+}}\over{\varepsilon}}=\theta$, then

[TABLE]

and

[TABLE]

[TABLE]

which implies that

[TABLE]

Thus (5.11) is changed into the form

[TABLE]

If we define

[TABLE]

which leads to

[TABLE]

then

[TABLE]

Now (5.11) gets by (5.10) the form

[TABLE]

Careful estimates will lead to a solution $z\in P^{2}(D^{\prime}_{+})$ and $z$ is quasi-periodic with the frequency $\omega$ in the first variable , which implies that $\phi$ is quasi-periodic with the frequency $\omega$ in the first variable and $\phi\in P^{2}(D^{\prime}_{+})$ can be determined.

2. Properties of $W$.

Since $H-\Omega$ is quasi-periodic with the frequency $\omega$ in the first variable and $|H-\Omega|_{D}\leq M,$ using Theorem 4.5 with $\rho={r\over 6}$, we get a solution $w$ which is quasi-periodic with the frequency $\omega$ in the first variable of (5.10) with

[TABLE]

Since $D^{\prime}_{+}\subseteq D(4\rho,t)$, we define $W=\Theta+w|_{D^{\prime}_{+}}$ and obtain $W-\Theta$ is quasi-periodic with the frequency $\omega$ in the first variable with

[TABLE]

which is the wanted estimate (5.6).

From (5.13) we also have

[TABLE]

An application of Lemma 5.1 with $d={2\over 3}s$ yields

[TABLE]

and which gets by means of (5.14) the form

[TABLE]

and consequently

[TABLE]

for all $z,z^{\prime}\in D(r^{\prime}_{+},{s\over 3})$. Inserting $z=\theta\xi,\ z^{\prime}=\theta\xi^{\prime}$ for $\xi,\xi^{\prime}\in D^{\prime}_{+}$, we get (5.7) because of

[TABLE]

Now we look for the range of $W$. Since

[TABLE]

then

[TABLE]

[TABLE]

[TABLE]

3. Estimate for $F_{2}=w\circ\Omega^{\star}-w\circ\Omega_{+}$.

First $\varepsilon<1$ because $\Gamma(\tau+1)\geq 1$ for $\tau>n$. Hence we get

[TABLE]

Since $\Omega^{\star}(D_{+})=D_{+},\ s_{+}<t=\theta^{-1}s,$ we have

[TABLE]

Moreover applying Lemma 5.1 to $\xi\mapsto w(\xi,\eta)$ with d=R,\ D=\big{\{}\xi\in\mathbb{C}:|\mathrm{Im}\ \xi|<4\rho\big{\}}, we obtain

[TABLE]

Using (5.15) and the definitions of $\Omega^{\star},\ \Omega_{+}$ yields

[TABLE]

By (5.15) and (I) we have

[TABLE]

hence

[TABLE]

4. Estimate for F_{3}=h\circ W\big{|}_{D_{+}}-h\circ\Theta.

Since \big{|}h\big{|}\leq M, we apply Lemma 5.1 with $d={2\over 3}s$ to obtain

[TABLE]

In the previous setting we have proved W(D_{+})\subseteq D-{2\over 3}s\subseteq D,\ \Theta(D_{+})\subseteq D-{2\over 3}s\subseteq D,\ then

[TABLE]

Moreover D_{+}\subseteq D^{\prime}_{+},\ \big{|}W-\Theta\big{|}_{D^{\prime}_{+}}\leq 2\varepsilon^{-1}M, hence

[TABLE]

Since q\leq 10^{-2}\theta^{2},\ by (I) we get

[TABLE]

5. Estimate for $F_{1}=w\circ\Omega_{+}-w\circ(\Omega_{+}+\phi)$.

Assume that

[TABLE]

In the previous setting we define

[TABLE]

By the definition of $h^{\star}$ we get \big{|}h^{\star}\circ\Theta\big{|}_{D_{+}}\leq M, and therefore

[TABLE]

Thus

[TABLE]

Since $s<{r\over 9}$, we have

[TABLE]

hence

[TABLE]

[TABLE]

From the definition of $r,\ s,\ \theta$, we have

[TABLE]

it is easy to see that

[TABLE]

Furthermore an application of Lemma 5.1 with $d={{2s}\over{3\theta}}$ yields

[TABLE]

As a consequence we have

[TABLE]

by (5.19). Since $q\leq 10^{-2}\theta^{2}$, we have

[TABLE]

6. Determination of $z$.

The set of all $z$ satisfying \big{|}z\big{|}_{{D_{+}}}\leq{1\over 24}\theta^{2}M is a complete metric space. Using (5.16), (5.17), (5.20), we get

[TABLE]

Hence

[TABLE]

is a mapping of this metric space into itself.

Furthermore letting

[TABLE]

we have as above

[TABLE]

for all $z,z^{\prime}$ satisfying (5.18), where $0<q<1$. Hence $F$ is a contraction, there is a fixed point $F(z)=z$, which leads to the existence of a mapping $\Phi$ such that the diagram in the inductive theorem exists and commutes. If the unknown function $z$ is represented by $\widetilde{{Z}}(\theta,\eta)$, where $z(\xi,\eta)=\widetilde{Z}(\omega\xi,\eta)$. From the above, we know $\widetilde{{Z}}$ is well defined in $D_{+}$, and by (5.21), $\widetilde{{Z}}$ have period $2\pi$ in each of variable $\theta_{i}(1\leq i\leq n).$ Hence, $z$ is quasi-periodic with the frequency $\omega$ in the first variable which means $\phi$ is quasi-periodic with the frequency $\omega$ in the first variable.

7. Proof of inequality (5.8).

Since \Phi_{+}-\Omega_{+}=\phi={\Theta^{-1}}(z+h^{\star}\circ\Theta),\ we know $\Phi_{+}-\Omega_{+}$ is quasi-periodic with the frequency $\omega$ in the first variable , as a consequence of

[TABLE]

we have the estimate

[TABLE]

thus we get

[TABLE]

The function

[TABLE]

is holomorphic for $|\eta|<\theta^{-1}s$, and has the estimate \Big{|}\theta^{-1}[g](\theta\eta)\Big{|}\leq\theta^{-1}M, since $g$ is the second component of $h$, and $|h(\theta\eta)|\leq M\ \text{for}\ |\eta|<\theta^{-1}s.$

We apply Lemma 5.2 to the function (5.23) with $q={\theta\over 2},\ m=3$ such that for the polynomial

[TABLE]

we have the estimate

[TABLE]

Then for

[TABLE]

we obtain $\Phi_{+}-\Omega_{+}-Q$ is quasi-periodic with the frequency $\omega$ in the first variable and

[TABLE]

Together with (5.22) we have

[TABLE]

The inequality (5.8) follows. The inductive theorem is proved. $\square$

Remark 5.4

Letting

[TABLE]

[TABLE]

and replacing the index $+$ by $k+1$, Theorem 5.1 confirms what we have asserted in Section 3 concerning the construction of the commuting diagram (3.12) observing (3.13), (3.14), (3.15) and (3.16).

6 Appendix

In this appendix we give the detail proof of Lemma 2.11 which have been used in the previous sections. For this purpose we need a well known and fundamental approximation result.

Lemma 6.1** (Lemma 2.1 in [1])**

Let $f\in\mathcal{C}^{p}(\mathbb{R}^{\ell})$ for some $p>0$ with finite $\mathcal{C}^{p}$ norm over $\mathbb{R}^{\ell}$. Let $\phi$ be a radial-symmetric, $\mathcal{C}^{\infty}$ function, having as support the closure of the unit ball centered at the origin, also $\phi$ is completely flat and takes value $1$, let $K=\hat{\phi}$ be its Fourier transform and for all $\delta>0$ define

[TABLE]

Then there exists a constant $c\geq 1$ depending only on $p$ and $\ell$ such that for any $\delta>0$, the function $f_{\delta}(x)$ is real analytic on $\mathbb{C}^{\ell}$, and for all $\beta\in\mathbb{N}^{\ell}$ with $|\beta|\leq p$, one has

[TABLE]

and, for all $0<\delta<\delta^{\prime}$,

[TABLE]

Moreover, the Hölder norms of $f_{\delta}$ satisfy, for all $0\leq s\leq p\leq r$,

[TABLE]

The function $f_{\delta}$ preserves periodicity, this is, if $f$ is T-periodic in any of its variable $x_{j}$, so is $f_{\delta}$.

Now we are in a position to prove Lemma 2.11.

**Proof of Lemma 2.11 ** Since the function $h(x,y)$ is quasi-periodic in $x$ with the frequency $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})$, from Definition 2.1, there exists the corresponding shell function

[TABLE]

which is $2\pi$-periodic in each $\theta_{j}$, such that $h(x,y)=F(\omega_{1}x,\omega_{2}x,\cdots,\omega_{n}x,y).$

From the assumptions of Lemma 2.11, $h\in\mathcal{C}^{p}(\mathbb{R}^{2})$, then $F\in\mathcal{C}^{p}(\mathbb{R}^{n+1})$, and \big{\|}F\big{\|}_{p} is equivalent to \big{\|}h\big{\|}_{p}. In fact, if $p\geq 0$ is an integer, then

[TABLE]

Similarly, if $p=l+s$,  $l\geq 0$ is an integer, $s\in(0,1),\nu:=(x,y),\bar{\nu}:=(\bar{x},\bar{y}),$ then

[TABLE]

where $\beta=(\beta_{1},\cdots,\beta_{n},\beta_{n+1})\in{\mathbb{N}}^{n+1},|\beta|=\beta_{1}+\cdots+\beta_{n}+\beta_{n+1},\lambda_{1}=\beta_{1}+\cdots+\beta_{n},\lambda_{2}=\beta_{n+1},\omega^{\lambda_{1}}={\omega_{1}}^{\beta_{1}}\cdots{\omega_{n}}^{\beta_{n}},\tilde{c}$ is a positive constant depending only on $p,\omega$.

An application of Lemma 6.1 to the function $F(\theta,y)\in\mathcal{C}^{p}(\mathbb{R}^{n+1})$, there exists a function $F_{\delta}$, which is an analytic approximation of $F$, and for any $\beta\in{\mathbb{N}}^{n+1}$ with $|\beta|\leq p$,

[TABLE]

and for any $0<\delta<\delta^{\prime},$

[TABLE]

where $c_{3}\geq 1$ is a positive constant depending only on $p$ and $n$.

Specially, if $\beta=0$, one has

[TABLE]

[TABLE]

where $0<\delta<\delta^{\prime}$. Hence,

[TABLE]

where $c_{4}$ is a positive constant depending only on $p,n.$

Now we define the analytic approximation of $h(x,y)$ in $E_{\delta}$ as follows

[TABLE]

Thus

[TABLE]

By (6), (6.2), we have

[TABLE]

Similarly, for any $0<\delta<\delta^{\prime},$ one can obtain

[TABLE]

where $c_{2}>0$ is a constant depending only on $p,n,\omega.$ Hence,

[TABLE]

The proof of this lemma is completed. $\square$