\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 159, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/159\hfil Gevrey-smoothness of invariant tori]
{Gevrey-smoothness of invariant tori for nearly integrable simplectic mappings}


\author[S. J. Jiang \hfil EJDE-2017/159\hfilneg]
{Shunjun Jiang}

\address{Shunjun Jiang \newline
College of Sciences,
Nanjing  Tech. University,
Nanjing, Jiangsu 210009, China}
\email{jiangshunjun@njtech.edu.cn}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted October 3, 2016. Published June 29, 2017.}
\subjclass[2010]{34C27, 37J40}
\keywords{Symplectic mappings;  KAM iteration; invariant tori;
\hfill\break\indent  non-degeneracy condition; Gevrey-smoothness}

\begin{abstract}
 In this article, we propose a  general normal form to prove the
 persistence and the Gevrey-smoothness of lower dimensional elliptic
 invariant  tori of nearly integrable symplectic mappings under
 the R\"ussmann  non-degeneracy condition. Our results generalize the
 ones presented in the literature.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

The KAM theory for nearly integrable Hamiltonian systems has been developed
extensively in the past decades 
(KAM theory was named after Andrey Kolmogorov, Vladimir Arnold and Jrgen Moser).
Studies under different non-degeneracy
conditions \cite{ar63,br,ko} generate various KAM theorems, among which
the non-degeneracy condition proposed by R\"ussmann  \cite{r1,r2}
sounds very useful and weaker.  In the KAM theory, the regularity of
KAM invariant tori is an important issue to consider, since small divisor
may usually cause the loss of smoothness. P\"oschel \cite{p5} proved that
the persisting invariant tori are C$^{\infty}$-smooth in the frequency parameter.
 Later Popov \cite{po} obtained the Gevrey-smoothness of invariant tori in
their frequencies under the Kolmogorov non-degeneracy condition.
Xu and You \cite{xu07}  extended this result to the case of the R\"ussmann
non-degeneracy condition.  Zhang and Xu \cite{zhang061,zhang062}
investigated the elliptic lower dimensional tori for Gevrey-smooth
Hamiltonian systems under R\"ussmann's non-degeneracy condition.

In addition to Hamiltonian systems, KAM theorems for mappings
\cite{ccq,De,DM,FDM,G,r1,xia,bi,lu} have been proven ever since Moser's well
 known work  \cite{m,mo} on area-preserving mappings. As we have seen, many
profound results for Hamiltonian systems can be generalized to symplectic
mappings since the latter are discrete Hamiltonian systems. This is one of
main motivations of our studies on the Gevrey-smoothness of elliptic
lower dimensional KAM invariant tori for symplectic mappings.

Despite the fact that some results in symplectic mappings can be extended
to Hamiltonian systems, there are still critical differences between
symplectic mappings and Hamiltonian systems. Normal form is crucial for
the study of the resonance relation between tangential frequencies and
normal frequencies. Unlike the normal form in Hamiltonian systems which
is unique, for symplectic mappings there is not a unique standard normal
form and in some cases is not easy to be discovered.
 Even if the normal form can
be discovered, it can cause much difficulty in the KAM iteration.
Moreover, symplectic mappings are determined implicitly by the generating
functions, which makes the KAM estimates more complicated.

Recently,  Lu et al \cite{lu} found a normal form for the elliptic lower
dimensional tori to prove the persistence of the invariant tori.
In this article, we provide a more generic normal form  to study the
persistence and Gevrey-smoothness of KAM tori, which is parameter-dependent
under the R\"ussmann non-degeneracy condition

Consider a family of  parameterized  symplectic mappings
 $$
\Phi: (x,u,y, v)\in \mathbb{T}^n\times \mathcal{W} \times \mathcal{O}
\times \mathcal{W} \to (\hat{x},\hat{u},\hat{y}, \hat{v})\in \mathbb{T}^n\times
\mathbb{R}^m\times \mathbb{R}^n\times \mathbb{R}^m,
$$
which   is implicitly defined by a generating function
 \begin{equation}
 H(x,u,\hat{y},\hat{v};\xi)= N+P,  \label{5261}
\end{equation}
with
\begin{equation}\begin{gathered}
\hat{x}=\partial_{\hat{y}}H(x,u, \hat{y},\hat{v};\xi),\quad
y =\partial_{x}H(x,u, \hat{y}, \hat{v};\xi),\\
\hat{u}=\partial_{\hat v}H(x,u, \hat{y},\hat v;\xi), \quad
v =\partial_{u}H(x,u, \hat{y},\hat{v};\xi),
\end{gathered}\label{1}
\end{equation}
 where
\begin{equation}
N(x, u, \hat{y},\hat{v};\xi)
=\langle x+\omega(\xi), \hat y\rangle+\langle Au,\hat v\rangle +\frac12\langle B u,
u \rangle+\frac12\langle C\hat v, \hat v\rangle,
\label{A1}
\end{equation}
and  $A,B,C$ are constant matrices.  We suppose that $ \xi \in\Pi$
is parameter and  $\Pi \subset  \mathbb{R}^n$ is a bounded closed
connected domain.

If $P=0$, $\Phi$ is expressed explicitly as
\begin{equation}\label{N2}
\begin{gathered}
\hat{x}=x+\omega(y),\quad \hat y= y, \\
\hat u=(A-C(A^{T})^{-1}B)u+C{(A^T)^{-1}} v,  \quad
\hat v=-{(A^T)^{-1}} B u+{(A^T)^{-1}} v .
\end{gathered}
\end{equation}
Let
\[
\Omega(A,B,C)=\begin{pmatrix}
A-C(A^T)^{-1}B & C(A^T)^{-1}\\
-(A^T)^{-1}B   &  (A^T)^{-1}
\end{pmatrix}_{2m\times 2m}.
\]
Then $ (\hat u,\hat v)^{T}=\Omega(A,B,C) (u,v)^{T}$. It is easy to see that
$ \mathbb{T}^{n}\times\{ 0,0,0\}$ is a lower dimensional invariant
torus with the rotational frequency $ \omega(\xi)$.

We call the  lower dimensional invariant torus to be elliptic if 1 is not
an eigenvalue of $ \Omega(A,B,C)$ and each  eigenvalue  has unit modulus;
 while hyperbolic if no eigenvalue   has unit modulus.
For simplicity,  let
 $$
A=\operatorname{diag}(a_1,a_2,\dots,a_{m}),\quad
B=\operatorname{diag}(b_1,b_2,\dots,b_{m}), \quad
C=\operatorname{diag}(c_1,c_2,\dots,c_{m}).$$

The rest of this article is organized as follows.
In Section 2, we present the related preliminary results on
the Gevery-class $G^{\mu}(\mathcal{O})$ of index $\mu\ (\mu\geq1)$
and state our main result. Section 3 is dedicated to the proof of our main result.
Section 4 is an appendix.

\section{Preliminaries}

Before stating our main results, we introduce some preliminary results on
assumptions, definitions and norm forms.
\begin{itemize}
\item[(H1)]  (Ellipticity condition)
 Suppose that $\Delta_{l}^{2}-4<0$, where
 $\Delta_{l}=\frac{a_{l}^{2}-b_{l}c_{l}+1}{a_{l}}$,
$ l=1,2,\dots, m$
\end{itemize}

 \begin{remark}\label{rm2.1} \rm
Direct calculations show that the eigenvalues of
$\Omega(A,B,C)$  are   $\frac{ \Delta_{l}\pm\sqrt{\Delta_{l}^{2}-4}}{2}$,
 $l=1,2,\dots,m$.   If  $\Delta_{l}^{2}-4<0$,
we have
$$
\big| \frac{ \Delta_{l}\pm\sqrt{\Delta_{l}^{2}-4}}{2} \big|=1.
$$
Let  $ \theta=(\theta_1,\theta_2,\dots,\theta_{m})$  such that
$e^{\pm \mathrm{i} \theta_{l}}=\frac{ \Delta_{l}\pm\sqrt{\Delta_{l}^{2}-4}}{2}$
and    $ 0< |\theta_{l}|\leqslant\frac{\pi}{2},~l=1,2,\dots,m$, where
$\mathrm{i}=\sqrt{-1}$
In this case, the   lower dimensional invariant torus is  elliptic.  We call
$ \theta$ the normal frequency.
If $ \Delta_{l}^{2}-4>0$, we have
$$
\big| \frac{ \Delta_{l}\pm\sqrt{\Delta_{l}^{2}-4}}{2} \big|
\neq 1,\; l=1,2,\dots,m.
$$
This means that the lower dimensional  invariant torus is hyperbolic.
 \end{remark}

  \begin{remark} \rm
If we choose $A,  B, C$ such that $ a_{i}=\sec\theta_{i}$,
$b_{i}=c_{i}=\tan\theta_{i}$,
 and $0<| \theta|\leqslant\frac{\pi}{2}$, the generated   function  \eqref{5261}
 reduces to the case described in \cite{lu}.
Note that the normal form in \cite{lu}   is unstable,
  which means that the normal form cannot remain after one KAM step, thus the
normalization is necessary at every KAM step in \cite{lu} .
  In this study, we use the above normal form, which can  persist under the
KAM iteration.
   \end{remark}
\begin{itemize}
\item[(H2)]  (R\"ussmann's non-degeneracy condition)  There exists an
integer $\bar{n}\geqslant 1$ such that
\begin{equation}\label{531}
\operatorname{rank}\{\partial^\beta_\xi \omega(\xi):
1\leqslant|\beta|\le \bar{n}\} =n,\quad \forall \xi\in \Pi.
\end{equation}
\end{itemize}

\begin{remark} \rm
The non-degeneracy condition \eqref{531} is slightly different from that
in Hamiltonian systems:
$$
\operatorname{rank}\big\{\partial^\beta_\xi \omega(\xi):
|\beta|\le \bar{n}\big\} =n,\quad \forall \xi\in \Pi.
$$
\end{remark}

\begin{itemize}
\item[(H3)] (Non-resonance  conditions)
 Suppose that for $ k\in Z^{n}$ with $|k|\neq 0$,
 $i,  j,  w \in \mathbb{Z}$ and $1 \leq i, j\leq   m$, $\omega(\xi)$ satisfies
\begin{gather}\label{s0}
|\langle k, \omega(\xi)\rangle -  2\pi w|
\ge \frac{2\alpha}{(2+|k|)^{\tau}}, \\
\label{s1} |\langle k, \omega(\xi)\rangle - \theta_i(\xi) - 2\pi w|\ge
\frac{2\alpha}{(2+|k|)^{\tau}}, \\
\label{s2}
 |\langle k, \omega(\xi)\rangle +\theta_i(\xi)\pm \theta_j(\xi)-2\pi w| \ge
\frac{2\alpha}{(2+|k|)^{\tau}},    ~~|k|+|i-j|\ne 0 .
\end{gather}
\end{itemize}

\begin{definition} \rm
 Let $\mathcal{O} \subset  \mathbb{R}^n$ be a bounded, closed, and
connected domain. A function $F: \mathcal{O}\to \mathbb{R}$ is
said to belong to the Gevery-class $G^{\mu}(\mathcal{O})$ of index
$\mu\ (\mu\geq1)$, provided that $F$ is $C^{\infty}(\mathcal{O})$-smooth and there
exists a constant $M$ such that for all $ p\in \mathcal{O}$, it holds
$$
| \partial^{\beta}_{p}F(p)|\leq cM^{|\beta|+1}\beta!^{\mu},
$$
where $|\beta|=\beta_1+\beta_2+\dots+\beta_{n}$ and
$\beta!^{\mu}=\beta_1!\beta_2!\dots\beta_{n}! $ for
$\beta=(\beta_1,\beta_2,\dots,\beta_{n})\in \mathbb{Z}^{n}_{+}$
\end{definition}

\begin{remark} \rm
From the definition, it is easy to see that the class $G^{1}$ of Gevery-smooth
functions coincides with the class of analytic functions, and it also satisfies
$$
G^{1}\subset G^{\mu_1} \subset G^{\mu_2} \subset C^{\infty},
$$
for $ 1<\mu_1<\mu_2<\infty$.
\end{remark}

Set
\begin{gather*}
\mathcal{T}_s=\{x \in \mathbb{C}^n/2\pi \mathbb{Z}^n:
|\operatorname{Im}x|_{\infty}\le s\}, \quad
 \mathcal{B}_r=\{ y\in \mathbb{C}^n: |y|_1\le r^2 \}, \\
\mathcal{W}_r=\{w\in \mathbb{C}^m:  ~|w|_2\le r\}.
\end{gather*}
Denote
\begin{gather*}
\mathcal{D}(s,r)=\mathcal{T}_s\times \mathcal{W}_r \times
\mathcal{B}_r\times \mathcal{W}_r,\\
|x|_{\infty}=\max_{1\le j\le n} |x_j|,\quad
|y|_1=\sum_{1\le j\le n}|y_j|, \quad |w|_2=\bigl(\sum_{1\le j\le m}
|w_j|^2\bigr)^{1/2}.
\end{gather*}
Let
\[
 \Pi=\{ \xi\in  \mathcal{O} :\operatorname{dist}(\xi,\partial
\mathcal{O} )\geqslant h\},\quad
\Pi_{h}=\{ \xi\in \mathbb{C}^n:\operatorname{dist}(\xi, \Pi)   \leqslant h\}.
\]

\begin{remark} \rm
By  definition, $ f \in G^{1,\mu}(\mathcal{D}(s,r)\times \Pi)$ which implies
 $ f(x,y,u,v;\xi) \in C^{\infty}(\mathcal{D}(s,r)\times \Pi) $ and
 $f(x,y,u,v;\xi) $ is analytic with respect to $(x,y,u,v) $ on $\mathcal{D}(s,r)$
and $ G^{\mu}$-smooth in $ \xi$ on $ \Pi_{h}$
\end{remark}

If  $P(x;\xi)$ is  analytic on $\mathcal{T}_{s}\times\Pi$, we can expand
$P(x;\xi)$ as the Fourier series
$$
P(x;\xi)=\sum_{k\in \mathbb{Z}^n}P_k(\xi)e^{\rm{i}\langle k,x\rangle}.
$$
We define
$$
\|P\|_s=\sum_{k\in \mathbb{Z}^n}|P_k|_{\Pi} e^{s|k|},\quad
|P_k|_{\Pi}=\max_{\xi\in \Pi}|P_k(\xi)|.
$$
When $P(x,u, \hat y,\hat v;\xi)$ is analytic on $\mathcal{D}(s,r)\times\Pi$, we let
$$
P(x,u, \hat y,\hat v;\xi)=\sum_{k\in \mathbb{Z}^n}
P_k( u, \hat y,\hat v;\xi)e^{\rm{i}\langle k,x\rangle},\quad
P_k(u, \hat y,\hat v;\xi)=\sum_{l,i,j}P_{klij}(\xi) \hat{y}^{l}u^{i}\hat{v}^{j}.$$
We define
$$
\|P\|_{D(s,r)\times\Pi}=\sum_{k\in \mathbb{Z}^n}|P_k |_{r} e^{s|k|},
$$
where
$$
|P_k |_{r}=\sup_{(u,\hat y, \hat v)\in \mathcal{W}_r\times \mathcal{B}_r\times
\mathcal{W}_r}\sum_{i,j,l}\|P_{klij}\|_s\hat y^l u^i{\hat v}^j.
$$
This norm is obviously  stronger than the sup-norm.
Moreover, the  Cauchy estimates of analytic functions are also valid
under this norm.
Let
$$
X_P=(-\partial_{\hat y}P, -\partial_{\hat v}P, \partial_{x}P, \partial_{u}P),
$$
endowed with the corresponding weighed norm
\begin{align*}
&\|X_P\|_{r;\mathcal{D}(s, r)\times\Pi}\\
&=\|\partial_{\hat y}P\|_{D(s,r)\times\Pi}+\frac{1}{r}\|\partial_{\hat
v}P\|_{D(s,r)\times\Pi}+\frac{1}{r^2}\|\partial_{x}P\|_{D(s,r)\times\Pi}
+\frac{1}{r}\|\partial_{u}P\|_{D(s,r)\times\Pi},
\end{align*}
where
\begin{gather*}
\|\partial_{\hat x}P\|_{D(s,r)\times\Pi}
=\sum  _j\|\partial_{\hat x_j}P\|_{D(s,r)\times\Pi},\quad
\|\partial_{\hat y}P\|_{D(s,r)\times\Pi}
=\max_j\|\partial_{\hat y_j}P\|_{D(s,r)\times\Pi},\\
\|\partial_{u}P\|_{D(s,r)\times\Pi}
=\Big(\sum_j(\|\partial_{u_j}P\|_{s,r})^2 \big)^{1/2}.
\end{gather*}

Now, we state our main result.

\begin{theorem}\label{thm2.7}
Consider the symplectic mapping $ \Phi(\cdot;\xi)$  defined by  \eqref{5261}.
Suppose that
$$
\tau\geqslant n \bar{n}-1,\quad \max_{\xi\in \Pi_{h}}
\big\{| \frac{\partial \omega(\xi)}{\partial \xi}|,
  |\frac{\partial \theta(\xi)}{\partial \xi}|\big\}  \leqslant T,
$$
and conditions (H1)--(H3) hold. There exists a $\gamma>0$ such that for any
$ 0<\alpha<1$, if
$$
\|X_P\|_{r;\mathcal{D}(s,r)\times\Pi_{h}}
=\epsilon\leqslant\gamma^{3} \alpha^{2\bar{\nu}}\rho^{2\nu},
$$
where
$\bar{\nu}=4(\bar{n}+1)$ and $\nu=4\tau(\bar{n}+1)+n+\bar{n}$,
 then the following two statements are true.

 (i) There exist a non-empty Cantor-like subset $\Pi_* \subset
\Pi$, parameterized symplectic
mappings $\Psi_{*}(\cdot; \xi)\in G^{1,\mu}(D(s/2,r/2)\times \Pi_{*})$,
and parameterized functions $H_{*}\in G^{1,\mu}(D(s/2,r/2)\times \Pi_{*})$
 such that
\begin{equation}
\label{542} \| \partial^{\beta}_{\xi}(\Psi_{*}-id)\|_{r;D(\frac{s}{2},
\frac{r}{2})\times\Pi_{*}}\leq  c\rho^{\nu}M^{|\beta|}
\beta!^{\mu}\gamma^{\frac{9}{4(n+1)}},\quad \forall \beta\in Z^{+}_{n}, \quad
\forall\xi\in \Pi_*,
\end{equation}
where $M= \frac{2T+1}{\alpha}[ \frac{4(\mu-1)(n+1)}{3}]^{\mu-1}$,
and $H_{*}(\cdot; \xi)=N_{*} +P_{*}$ satisfies
\begin{gather*}
N_{*}(x, u, \hat y, \hat v;\xi)=\langle x+\omega_{*},\hat y\rangle +
 \langle A_{*}u, \hat v\rangle+\frac12 \langle B_{*}u, u\rangle
+\frac12 \langle C_{*}\hat v, \hat v\rangle,\\
P_{*}( x, u,\hat y, \hat v; \xi)=\sum_{|i|+|j|+2|l|
\ge3}P_{lij}(x;\xi)\hat y^lu^i\hat v^j.
\end{gather*}
Moreover,  $\Phi_{*}(\cdot; \xi)=\Psi_{*}^{-1}\circ
\Phi\circ\Psi_{*}$ is generated by $H_{*}(\cdot; \xi)=N_{*}
+P_{*}$.

(ii) For $\xi\in \Pi_*$,
the symplectic mapping $\Phi(\cdot; \xi)$ admits an invariant torus
$$
\{ T_{\xi}=\Psi_{*}(T^n,0,0,0;\xi): \xi\in \Pi_*\}
$$
whose tangential  frequency $ \omega_{*}$  and normal frequency
$ \theta_{*}$ satisfy
\begin{gather}\label{543}
 | \partial^{\beta}_{\xi}(\omega_{*}(\xi)-\omega(\xi))|_{\Pi_{*}}
\leq c\rho^{2\nu}M^{|\beta|}\beta!^{\mu}\gamma^{\frac{9}{4(n+1)}}, \\
\label{9.118}
 | \partial^{\beta}_{\xi}(\theta_{*}(\xi)-\theta(\xi))|_{\Pi_{*}}\leq c\rho^{2\nu}M^{|\beta|}\beta!^{\mu}\gamma^{\frac{9}{4(n+1)}}.
 \end{gather}
Moreover,  for $i,j \in \mathbb{Z}$ and $1\leq i,j\leq   m$, we have
\begin{equation}\label{9.119}
  |\langle \omega_{*}(\xi),k\rangle -s_1\theta_{*i}(\xi)
- s_2\theta_{*j}(\xi)-2\pi w|\geq \frac{\alpha}{(2+|k|)^{\tau}},
\end{equation}
where  $\xi \in \Pi_{*}$, $0 \neq k\in Z^{n}$,
$0\leqslant |s_1|+|s_2|\leqslant 2$, and $s_{d}\in \mathbb{Z}$ $(d=1,2)$.
 In addition, we have
$$
 \operatorname{meas} (\Pi\setminus
\Pi_*)\to 0, \quad\text{as } \alpha\to 0.
$$
\end{theorem}


\section{Proof of main result}

\subsection{KAM-steps}

To prove our main result, we apply the idea for Hamiltonian  systems
\cite{p2,xu07} as well as some technical lemmas.
\medskip

\noindent\textbf{KAM iteration lemma:}
 For the symplectic mapping $\Phi(\cdot;\xi)$ defined by \eqref{5261},
when $ \delta\in (0,1)$, let $ \mu=\tau+\delta+2$,
$ \sigma=(\frac{3}{4})^{\frac{\delta}{\tau+1+\delta}}$, $
0<E<1$, $0<\eta<\frac{1}{8}$ and $0<\rho=(1-\sigma)s/10<\frac{s}{5}$.
  Let
  $$
\max_{\xi\in \Pi_{h}} \big\{| \frac{\partial \omega(\xi)}{\partial \xi}|,
 |\frac{\partial \theta(\xi)}{\partial \xi} |\big\}  \leqslant T,
\quad  h=\frac{\alpha}{(2+K)^{\tau+1}T},
$$
where  $K>0$ satisfies $\eta^{2}e^{-K\rho}=E$.
Suppose that conditions (H1)--(H3) hold and $P$ satisfies
$$
\|X_P\|_{r;D(s,r)\times\Pi_{d}}\leqslant  \epsilon
=\eta^{2}\alpha^{2\bar{\nu}} \rho^{2\nu}E
$$
with $0<\alpha<1$, $\bar{\nu}=4(\bar{n}+1)$ and
$\nu=4\tau(\bar{n}+1)+n+\bar{n}$.
 Then the following three statements are true.

(i) For $ \xi \in \Pi_{h}$, there exists a symplectic
 diffeomorphism $ \Psi(\cdot;\xi)$   with
$$
  \| \Psi-id\|_{r;D(s-3\rho, \frac{r}{4})\times\Pi_{h}}
\leq \frac{c \epsilon }{\alpha^{\bar{\nu}} \rho^{\nu}}, \quad
\| D\Psi-id\|_{r;D(s-3\rho, \frac{r}{4})\times\Pi_{h}}
\leq \frac{c \epsilon }{\alpha^{\bar{\nu}} \rho^{\nu+1}},
$$
such that the conjugate mapping $ \Phi_{+}(\cdot;\xi)=\Psi^{-1}\circ \Phi\circ\Psi$
 is generated by $ H_{+}(\cdot;\xi)=N_{+}+P_{+}$, where
$$
N_{+}=\langle x+\omega_{+}(\xi),\hat{y}\rangle+\langle A_{
 +}u,\hat{v}\rangle+\frac{1}{2}\langle B_{
 +}u,u\rangle+\frac{1}{2}\langle C_{
 +}\hat{v},\hat{v}\rangle
  $$
and $P_{+}$
    satisfies
$$
\|X_{P}\|_{r_{+};D(s_{+},r_{+})\times\Pi_{d}}
\leqslant\eta_{+}^{2} \alpha^{2\bar{\nu}}_{+}\rho^{\nu}_{+}E_{+}= \epsilon_{+}
$$
 with
$$
s_{+}=s-5\rho,\quad \rho_{+}=\sigma\rho,\quad \eta=E,\quad
 r_{+}=\eta r,\quad E_{+}=E^{\frac{4}{3}},\quad
 \frac{\alpha }{2} \leqslant \alpha_{+} \leqslant \alpha.
$$
Let $  e^{\pm \mathrm{i}\theta_{+l}}$ be the eigenvalues of
$\Omega(A_{+},B_{+},C_{+})$, where
$ \theta_{+}=(\theta_{+1},\theta_{+2},\dots,\theta_{+m} )$ and $l=1,2,\dots,m$.
We have
 \begin{equation}\label{4301}
 |\omega_{+}(\xi)-\omega(\xi)|\leq \epsilon,\quad
|\theta_{+}(\xi)-\theta(\xi)|\leq c\epsilon,\quad \forall \xi \in \Pi_{h}.
\end{equation}

(ii) Let
$\alpha_+=\alpha-(K+2)^{\tau+1}\epsilon$,
\begin{align*}
\bar{\Pi}=\Big\{& \xi\in  \Pi:
|\langle \omega_{+}(\xi),k\rangle -s_1\theta_{+i}(\xi)- s_2\theta_{+j}(\xi)
-2\pi w|< \frac{2\alpha_{+}}{(2+|k|)^{\tau}},\ k\in Z^{n},\\
& K<|k|\leq K_{+},\; 0\leqslant |s_1|+|s_2|\leqslant 2,\;
 s_{d}\in \mathbb{Z}\ (d=1,2)\Big\},
\end{align*}
 and $ \Pi_{+}=\Pi\setminus \bar{ \Pi}$.
Then for $\xi\in \Pi_{+}, \ \forall k\in Z^{n}$  and $0<|k|\leq K_{+}$, we have
\begin{equation}
| \langle \omega_{+}(\xi),k\rangle -s_1\theta_{+i}- s_2\theta_{+j}
-2\pi w|\geqslant \frac{2\alpha_{+}}{(2+|k|)^{\tau}},
 \end{equation}
where $K_{+}>0$ satisfies $\frac{ e^{-K_{+}\rho_{+}}}{\eta_{+}^{2}}= E_{+}$.

(iii) Let $T_{+}=T+\frac{6\epsilon}{h}$ and
$h_{+}=\frac{\alpha_{+}}{2(K_{+}+2)^{\tau+1}T_{+}}$.
If $h_{+}\leq \frac{5}{6}h$, we have
$$
\max_{\xi\in \Pi_{h_{+}}}
\big\{| \frac{\partial \omega_{+}(\xi)}{\partial \xi}|,\;
|\frac{\partial \theta_{+}(\xi)}{\partial \xi}|  \big\} \leq T_{+},
$$
where $\Pi_{h_{+}}$ is the complex $h_{+}$-neighborhood of $\Pi_{+}$
\smallskip

\noindent\textbf{A. Generating functions of conjugate mappings:}
Let $p=(x,u)$ and $q=(y,v)$.   The  symplectic structure
becomes $dp\wedge dq$  on $\mathbb{R}^{n+m}\times
\mathbb{R}^{n+m}$. Consider a symplectic mapping
$\Phi: (p, q)\to (\hat{p},\hat{q})$ generated by
\begin{equation}
\hat
p=\partial_{\hat q} H(p,\hat q)= H_2(p,\hat q)\quad\text{and}\quad
q=\partial_{p} H(p,\hat )= H_1(p,\hat q). \label{A*}
\end{equation}
 The generating function is
 $H(p,\hat q)= N(p,\hat q)+P(p, \hat q)$,
 where $N$ represents the main term
 and $P$ is a small perturbation.
Define a symplectic  transformation  $\Psi: (p_+, q_+)\to (p, q)$ by
\begin{equation}\label{t1}
q=q_++F_1(p,q_+)\quad \text{and}\quad p_+=p+F_2(p,q_+).
\end{equation}
  The generating function is $\langle p, q_+\rangle +F(p, q_+)$
with $F$ being a  small  function.
So  $\Psi $  approaches to the identity.
Then, we get a conjugate mapping
$$
\Phi_{+}=\Psi^{-1}\circ \Phi\circ \Psi: (p_+,q_+)\to (\hat{p}_+,\hat{q}_+)
$$
implicitly by
\begin{equation}\label{A**}
\hat p_+=H_2(p,\hat q)+F_2(\hat p, {\hat q}_+) \quad\text{and}\quad
 q_+=H_1(p,\hat q)-F_1(p, q_+).
\end{equation}

From the following Lemma, $\Phi_{+}$ is generated by  a  function
$H_{+}(p_{+}, \hat{q}_{+})$.

\begin{lemma}[\cite{lu}]\label{l4}
 The conjugate symplectic mapping  $\Phi_{+}$ can be  determined  by
$H_{+}(p_+, {\hat q}_+)$ through
\begin{equation}\label{A***}
\hat p_+=\partial_{\hat q_+} {H_+}(p_+,\hat
q_+), \ q_+=\partial_{p_+} {H_+}(p_+,\hat q_+),
\end{equation}
where
\begin{equation}
\begin{aligned}\label{t3}
  H_{+}(p_+, {\hat q}_+)
&=H(p,\hat q)+H_1(p,\hat q)F_2(p,q_+)-H_2(p,\hat q)F_1(\hat p, {\hat q}_+)\\
&\quad +F(\hat p, {\hat q}_+)-F(p,q_+)-F_1(p,q_+)F_2(p,q_+),
\end{aligned}
\end{equation}
with  $p,\hat p, \hat q, q_+$ depending on $(p_+,\hat q_+)$ as explained  above.
Moreover,  if we set $ z=(p_+, \hat q_+)$, then  we have
\begin{equation}
\label{H+}
 H_{+}(z)=H(z)+F(N_2(z),\hat q_+)-F(p_+, N_1(z))+Q(z).
\end{equation}
The small term $Q(z) $ has the estimate
\begin{equation}\label{8.106}
\|X_{Q}\|_{r;\mathcal{D}(s-5\rho, r/16)\times\Pi}\le
\frac{c\epsilon^2}{\alpha^{2\bar{\nu}}\rho^{2\nu}},
\end{equation}
with $\bar{\nu}= 4(\bar{n}+1)$ and $\nu=\bar{n}+n+4\tau(\bar{n}+1)$.
\end{lemma}


\noindent\textbf{B. Truncation:}
Let
 \begin{equation}\label{8.105}
 P=R+ (P-R),
 \end{equation}
 where
\begin{equation}\label{R}
\begin{aligned}
 R(p, \hat q)&=P_{000}(x)+\langle P_{100}(x), \hat y\rangle +\langle P_{010}(x),
u\rangle +\langle P_{001}(x), \hat v\rangle \\
&\quad + \langle P_{011}(x)u,  \hat v\rangle +\frac12 \langle
P_{020}(x)u, u\rangle +\frac12 \langle P_{002}(x)\hat v, \hat
v\rangle,\end{aligned}
\end{equation}
with
$$
P_{lij}=\frac{\partial^{l+i+j} P}{\partial {\hat y}^l \partial u^i
\partial {\hat v}^j }|_{ u=0, \hat y=0, \hat v=0}, \quad  2|l|+|i|+|j|\le 2.
$$
So we have
$$
P-R=\sum_{2|l|+|i|+|j|\leqslant 2,k\geqslant K}P_{lij}\hat{y}^{l}u^{i}
\hat{v}^{j}+\sum_{2|l|+|i|+|j|\geqslant 3}P_{lij}\hat{y}^{l}u^{i}\hat{v}^{j}.
$$
\smallskip

\noindent\textbf{C. Extension of small divisor estimate:}
For $  \xi \in \Pi_{h}$, there exists a $ \xi_0\in\Pi$ such that
$ | \xi-\xi_0|<h$ For  $ |k|\leq K$, we have
\begin{equation}\label{e3.12}
\begin{split}
& |\langle \omega(\xi)-\omega(\xi_0),k\rangle +s_1(\theta_{i}(\xi)-\theta_{i}(\xi_0)) +s_2(\theta_j(\xi)-\theta_j(\xi_0)) | \\
&\leqslant  |\langle \omega(\xi)-\omega(\xi_0),k\rangle |+|s_1\|(\theta_{i}(\xi)-\theta_{i}(\xi_0))| +|s_2\|(\theta_j(\xi)-\theta_j(\xi_0)) |\\
&\leqslant  (k+|s_1|+|s_2|) T h \\
&\leqslant  (k+2)Th \\
&\leqslant  \frac{\alpha}{(K+2)^{\tau}}.
\end{split}
\end{equation}
It follows from \eqref{s0}--\eqref{s2} and  \eqref{e3.12} that
 \begin{equation}\label{9.142}
|\langle \omega(\xi),k\rangle  +s_1\theta_{i}(\xi)
+s_2\theta_j(\xi)-2\pi w| \geqslant \frac{\alpha}{(2+|k|)^{\tau}},
 \end{equation}
 where  $ h=\frac{\alpha}{(2+K)^{\tau+1}T}$,
$0\leqslant |s_1|+|s_2|\leqslant 2$ and $s_{d}\in \mathbb{Z}\ (d=1,2)$\,.
\smallskip

\noindent\textbf{D. Homological equations:}

Following the idea described in \cite{p2}, we consider the homological equation:
 $$
 N(p_+,\hat q_+)+R( p_+,\hat q_+)-F(p_+, N_p(p_+,\hat q_+))
+F(N_q(p_+,\hat q_+),\hat q_+)=\bar{N}(p_+,\hat q_+),
$$
where  $F(p, \hat q)$ possess the same form as \eqref{R}.
Just for simplicity, here and below we drop the subscripts `$+$'
in $p_+$ and $\hat q_+$.

 Let  $x+\omega=\tilde x $. Denoting
 $$
\hat{p}=N_{\hat{q}}(p, \hat{p})=(\tilde x , Au+Cv), \quad
q=N_{p}(p, \hat{p})=(\hat{y},A\hat{v}+Bu),
$$
we have
$$
F(N_q(p,\hat q),\hat q)-F(p, N_p(p,\hat q))=L_0+L_1+L_2,
$$
where $L_0,L_1,L_2 $ indicate  the $ith$ ($i=0,  1,  2)$ order terms of $u$
and $ \hat v$ respectively:
\begin{gather*}
L_0=(F_{000}(\tilde x )- F_{000}(x ))+   \langle  F_{100}(\tilde x)-
F_{100}(x), \hat y\rangle , \\
 L_1=\langle A^TF_{010}(\tilde x)-F_{010}(x)-BF_{001}(x), u\rangle
+\langle CF_{010}(\tilde x)+F_{001}(\tilde x)-AF_{001}(x), v\rangle,\\
\begin{aligned}
L_2&= \langle \{F_{011}(\tilde x)A  - AF_{011}(x)+CF_{020}(\tilde
x)A-AF_{002}(x) B\}u, \hat v\rangle \\
&\quad  +\frac12 \langle \{A^T F_{020}(\tilde x)A
 -  F_{020}(x) -B F_{002}(x) B-BF_{011}(x)-F_{011}^T(x) B\}u, u\rangle \\
&\quad + \frac12 \langle \{C F_{020}(\tilde x)C
 +  F_{002}(\tilde x) -AF_{002}(x)A^T+F_{011}(\tilde x) C
 +CF_{011}^T(\tilde x)\}\hat v, \hat v\rangle.
\end{aligned}
\end{gather*}
We consider the  equations
\begin{equation}\label{8.101}
\begin{gathered}
L_0=(R_{000}(x)-[R_{000}])+  \langle R_{100}(x)-[R_{100}] , \hat y \rangle,\\
L_1=\langle R_{010}(x), u \rangle+ \langle R_{001}(x), \hat v \rangle,\\
L_2=\langle (R_{011}(x)-\hat A)u, \hat v \rangle
 + \frac{1}{2} \langle (R_{020}(x)-\hat B)u,u \rangle
 + \frac{1}{2}\langle (R_{002}(x)-\hat C)\hat v, \hat v \rangle,
\end{gathered}
\end{equation}
where $\hat A,  \hat B$ and $\hat C$ are to be determined.

We start with the equation
$$
F_{j00}(x+\omega)-F_{j00}(x)=R_{j00}(x)-[R_{j00}],\quad j=0, 1,
$$
by expanding $F_{j00}(x)$ and $R_{j00}(x)$ as the Fourier series:
$$
F_{j00}(x)=\sum_{k\in \mathbb{Z}^n} F_{kj00}e^{\rm{i}\langle k, x\rangle},\quad
R_{j00}(x)=\sum_{k\in \mathbb{Z}^n} R_{kj00}e^{\rm{i}\langle k, x\rangle}.
$$
 It follows that
 \begin{equation}
\label{8.102}
F_{kj00}=\frac{1}{e_k-1} R_{kj00},
\end{equation}
with $e_k=e^{\rm{i}\langle k\ (\omega\rangle}, k\ne 0)$
By \eqref{9.142}, we have the estimate
 \begin{equation}
\label{9.151}
\|F_{j00}\|_{(s-\rho)\times\Pi }
\le \frac{c\|R_{j00}\|_s}{\alpha^{\bar{n}+1}\rho^{\bar{n}+n+\tau(\bar{n}+1)}}.
 \end{equation}

Next we solve the second equation of \eqref{8.101}.
Let $F_{010}=(F^1_{010}, \ldots, F^m_{010})$  and
$F_{001}=(F^1_{001}, \ldots, F^m_{001})$
and expand $F^{l}_{0i'j'}(x)$ and $R^{l}_{0i'j'}(x)$ as the Fourier series:
$$
F^{l}_{0i'j'}(x)=\sum_{k\in \mathbb{Z}^n} F^{l}_{k0i'j'}e^{\rm{i}\langle k, x\rangle},
\quad
R^{l}_{0i'j'}(x)=\sum_{k\in \mathbb{Z}^n} R^{l}_{k0i'j'}e^{\rm{i}\langle k, x\rangle}
$$
with $ l=1,2,\dots,m$ and $(i',j')=(0,1)$ or $(1,0)$

By the definition of $L_1$ and the second equation of \eqref{8.101},
one can see the relation between $F^{l}_{0i'j'}(x)$ and $R^{l}_{0i'j'}(x)$:
 \[
M_{l} \cdot \begin{pmatrix}   F^{l}_{k010}( x)  \\
F^{l}_{k001}(x)
\end{pmatrix}
=\begin{pmatrix} R^{l}_{k010} \\
R^{l}_{k001}
\end{pmatrix},
\]
where
  \[M_{l}=\begin{pmatrix} a_{l}e_k-1 & -b_{l} \\
c_{l}e_k& e_k-a_{l}
\end{pmatrix}
\]
with $e_{k}=e^{\rm{i}\langle k, \omega\rangle}$.
By a straightforward calculation, we have
\begin{align*}
\det(M_{l})&=\Big(e_{k}- \frac{ \Delta_{l}+\sqrt{\Delta_{l}^{2}-4}}{2}\Big)
\Big(e_{k}- \frac{ \Delta_{l}-\sqrt{\Delta_{l}^{2}-4}}{2}\Big) \\
&=-2\Big(\sin\frac{\langle k, \omega\rangle+\theta_{l}}{2}
 -\mathrm{i}\cos\frac{\langle k, \omega\rangle+\theta_{l}}{2}\Big)
 \sin\frac{\langle k, \omega\rangle-\theta_{l}}{2},
\end{align*}
where $\theta_{l}$, $\Delta_{l}$ $(l=i,j)$, are defined in Remark \ref{rm2.1}.
By  \eqref{9.142}
we know $ |\det(M_{l})|\geqslant  \frac{\alpha^{2}}{(2+|k|)^{2\tau}}$.
Note that
$$
F^{l}_{k0i'j'}=    \frac{\tilde{R}^{l}_{i'j'}}{ |\det(M_{l})|}
$$
with  $ \tilde{R}^{l}_{i'j'}=c_1R^{l}_{k010}(x)+c_2R^{l}_{k001}(x)$ Then
 \begin{equation}
\label{9.152}
 \|F_{0i'j'}\|_{D(s-\rho,r)\times\Pi}\leq
\frac{c\|R_{0i'j'}\|_s }{\alpha^{2\bar{n}+2}\rho^{2\tau(\bar{n}+1)+\bar{n}+n}}
 \end{equation}
with $ (i',j')=(0,1)$ or $(1,0)$.

Before solving the third equation of \eqref{8.101},
 let us consider the equation
\begin{equation}
 L_2= \langle R_{011}(x)u, \hat v \rangle+ \frac{1}{2} \langle R_{020}u, u \rangle+
 \frac{1}{2} \langle R_{002} \hat v, \hat v \rangle. \label{811}
 \end{equation}
Let $F_{0\,i'j'}=(F_{0\,i'j'}^{ij})_{1\le i,j \le m}$ with $(i',
j')=(1,1),  (2,0)$ or $ (0,2)$
We expand $F_{0\, i'j'}^{ij}$ and $ R_{0\,i'j'}^{ij}$ as
$$
F_{0\,i'j'}^{ij}=\sum_{k\in \mathbb{Z}^n}F_{k0\ i'j'}e^{\rm{i}\langle
k,x\rangle},\quad
 R_{0\,i'j'}^{ij}=\sum_{k\in \mathbb{Z}^n}R_{k0\, i'j'}e^{\rm{i}\langle
k,x\rangle}.
$$
From the definition of $L_2$ and  \eqref{811}, we have
\[
N_{ij} \begin{pmatrix} F_{k011}^{ji} \\
  F_{k011}^{ij} \\ F_{k020}^{ij}
   \\
F_{k002}^{ij}
\end{pmatrix} =\begin{pmatrix} R_{k011}^{ji} \\
  R_{k011}^{ij} \\ R_{k020}^{ij}
   \\
R_{k002}^{ij}
\end{pmatrix},
\]
where
\[
N_{ij}=\begin{pmatrix}
0 & e_ka_j-a_i & e_kc_ia_j & -a_ib_j \\
  e_ka_i-a_j & 0 & e_ka_ic_j & -b_ia_j \\ -b_j & -b_i & e_ka_ia_j-1
  & -b_ib_j \\
e_kc_i & e_kc_j & e_kc_ic_j
  & e_k-a_ia_j
\end{pmatrix}.
\]
A direct calculation gives
$\det(N_{ij})=S_{4}e_{k}^{4}+S_{3}e_{k}^{3}+S_2e_{k}^{2}+S_1e_{k}+S_0$,
where
\begin{gather*}
S_{4}= a_{i}^{2}a_j^{2},\quad
S_0=a_{i}^{2}a_j^{2},\\
\begin{aligned}
S_{3}&= S_1=a_{i}^{3}a_j^{3}-a_{i}^{3}a_jb_jc_j-a_j^{3}a_{i}b_{i}c_{i}
 +a_ja_{i}b_jb_{i}c_jc_{i},\\
&\quad +a_ja_{i}^{3}+a_j^{3}a_{i}-a_ja_{i}b_{i}c_{i}  -a_ja_{i}b_jc_j
  +a_ja_{i},
\end{aligned} \\
\begin{aligned}
S_2&= a_{i}^{4}a_j^{2} -a_{i}^{2}a_j^{4} +2  a_j^{2}a_{i}^{2}b_{i}c_{i}
  +2 a_j^{2}a_{i}^{2}b_jc_j
  - a_{i}^{2}b_j^{2} c_j^{2}   \\
& \quad -   a_j^{2}b_{i}^{2} c_{i}^{2}    -2 a_{i}^{2}a_j^{2}+2 a_{i}^{2}b_j c_j
  +   2 a_j^{2}b_{i} c_{i}-a_{i}^{2}  -a_j^{2}.
 \end{aligned}
\end{gather*}
For $i, j=1,2,\dots,m$, we find
$$
\det( N_{ij})=\left(e_{k}-e^{\mathrm{i}\theta_{i}}
e^{\mathrm{i}\theta_j}\right)\left(e_{k}-e^{\mathrm{-i}\theta_{i}}
e^{\mathrm{-i}\theta_j}\right)\left(e_{k}-e^{\mathrm{i}\theta_{i}}
e^{\mathrm{-i}\theta_j}\right)\left(e_{k}-e^{\mathrm{-i}\theta_{i}}
e^{\mathrm{i}\theta_j}\right),
$$
with  $ \theta_{l}\ (l=i,j)$  given as  in Remark \ref{rm2.1}.
By  \eqref{9.142},
we have
$$
|\det(N_{ij})|\geqslant  \frac{\alpha^{4}}{(2+|k|)^{4\tau}},
$$
with $|k|+|i-j|\neq 0$.
Thus we can solve the equation \eqref{811} in the case of $|k|+|i-j|\neq 0$ and get
 \begin{equation}
\label{8.104}
F_{k0i'j'}^{ij} = \frac{\tilde{R}^{ij}}{|\det(N_{ij})|},
\end{equation}
 with $ \tilde{R}^{ij}= c_1R_{k011}^{ji}
 +c_2 R_{k011}^{ij} +c_{3} R_{k020}^{ij}   +c_{4}
R_{k002}^{ij}$.

From \eqref{811} and \eqref{8.104}, we consider the third equation of
\eqref{8.101} by setting
\begin{equation}\label{AB}
\begin{gathered}
 \hat\omega=\operatorname{diag}(\hat\omega_1, \ldots, \hat\omega_n),\quad
 \hat A=\operatorname{diag}( \hat A_1, \ldots,  \hat A_m),\\
\hat B=\operatorname{diag}(\hat B_1, \ldots,  \hat B_m), \quad
\hat C=\operatorname{diag}( \hat C_1, \ldots, \hat C_m),
\end{gathered}
\end{equation}
with
$$
\hat\omega_j= [R_{100}^{jj}],\quad \hat A_j= [R_{011}^{jj}],\quad
\hat B_j= [R_{020}^{jj}],\quad \hat C_j= [R_{002}^{jj}].
$$
By a similar discussion as the above, one can deduce that
 \begin{equation} \label{9.153}
 \|{F_{0i'j'}}\|_{D(s-\rho,r)\times\Pi}\leq
\frac{c\|R_{0i'j'}\|_s }{\alpha^{4\bar{n}+4}\rho^{4\tau(\bar{n}+1)+\bar{n}+n}}
 \end{equation}
with $ (i',j')=(1,1)$,  $(2,0)$ or $(0,2)$

It follows from \eqref{9.151}, \eqref{9.152} and \eqref{9.153} that
  \begin{equation}
\label{9.154}
 \|X_{F}\|_{r;D(s-\rho,r)\times\Pi}\leq
 \frac{c\epsilon }{\alpha^{\bar{\nu}}\rho^{\nu}}
  \end{equation}
 with $\bar{\nu}=4(\bar{n}+1)$ and $\nu=4\tau(\bar{n}+1)+n+\bar{n}$.

Let $\chi: (p, q)\to (-F_{y_+},  F_x)$
Since  $ \Psi=id+\chi$, we combine the estimate of $ F$ in \eqref{9.154}
 and the Cauchy estimate  to obtain
\begin{gather*}
\| \Psi-id\|_{r;D(s-3\rho, \frac{r}{4})\times \Pi}
\leq \frac{c\epsilon }{\alpha^{\bar{\nu}}\rho^{\nu}}, \\
\| D\Psi-id\|_{r;D(s-3\rho, \frac{r}{4})\times \Pi}
\leq  \frac{c\epsilon }{\alpha^{\bar{\nu}}\rho^{\nu+1}}.
\end{gather*}
\smallskip


\noindent\textbf{E. Choices of parameters in KAM iteration:}
Set
$$
0<E<1,\quad  \eta=E,\quad \epsilon=\eta^{2}\alpha^{2\overline{\nu}}\rho^{2\nu}E,\quad
\frac{e^{-K\rho}}{\eta^{2}}=E,\quad h=\frac{\alpha}{2(K+2)^{\tau+1}T}.
$$
Let $ \sigma\in(0,1)$ We denote
\begin{gather*}
\rho_+=\sigma\rho,\quad s_+=s-5\rho,\quad r_+=\eta r,\\
\alpha_+=\alpha-(K+2)^{\tau+1}\epsilon,\quad \epsilon_+=c\eta \epsilon, \quad
 E_+=cE^{\frac{4}{3}}.
\end{gather*}
From the equality
$$
P-R=\sum_{|2l|+|i|+|j|\leqslant 2, k\geqslant K}P_{lij}\hat{y}^{l}u^{i}\hat{v}^{j}
+\sum_{2|l|+|i|+|j|\geqslant 3}P_{lij}\hat{y}^{l}u^{i}\hat{v}^{j},
$$
we get
\begin{equation}\label{8.241}
\|X_{P-R}\|_{\eta r; \mathcal{D}(s-5\rho, \eta r)\times\Pi}
\leqslant  c\cdot\epsilon \Big( \eta+\frac{e^{-K\rho}}{\eta^{2}} \Big).
\end{equation}
By \eqref{8.106} and \eqref{8.241}, we have
 \begin{align*}
 \|X_{P_+}\|_{\eta r; D(s-5\rho, \eta r)\times\Pi_{+}}
&\le c\cdot\epsilon\Big(\eta+\frac{e^{-K\rho}}{\eta^{2}}\Big) +
\frac{c\epsilon^2}{\eta^{2}\alpha^{2\bar \nu}\rho^{2\nu}}\\
&\leq  c\eta \epsilon=c\alpha^{2\overline{\nu}}\rho^{2\nu}E^{4}\\
&\leq \alpha^{2\bar{\nu}}_+\rho^{2\nu}_+E_+^{3}.
\end{align*}
Setting $ \epsilon_+=\alpha^{2\bar{\nu}}_+\rho^{2\nu}_+E_+^{3}$,
so we arrive at
$$
\|X_{P_+}\|_{r_+;\mathcal{D}(s_+,r_+)\times\Pi_{+}}\le \epsilon_+,
$$
Given the  choice of $ \alpha_+$,
for $\xi\in \Pi_{+}$ and $ 0\neq k\leq K$, we have
\begin{align*}
|\langle k,\omega_+(\xi) \rangle -2\pi w| \\
&\geqslant|\langle k,\omega(\xi)\rangle
 +2\pi w|-|\langle k,\omega_{+}(\xi)-\omega(\xi) \rangle | \\
&\geq \frac{2}{(2+|k|)^{\tau}}[\alpha-(2+K)^{\tau+1}\epsilon].
\end{align*}
Similarly, for sufficiently large $K$  we have
$$
|\langle k,\omega_+(\xi) \rangle  +s_1\theta_{+i}(\xi)
+s_1\theta_{+j}(\xi)-2\pi w|\geqslant \frac{2}{(2+|k|)^{\tau}}
[\alpha-(2+K)^{\tau+1}\epsilon],
$$
with $ 0<|s_1|+|s_2|\leqslant 2,\ s_{d}\in \mathbb{Z,}\ (d=1,2)$,
$\xi\in \Pi_{+}$ and $ 0\neq k\leq K$
In view of $\alpha_{+}= \alpha-(2+K)^{\tau+1}\epsilon$, we have
$$
| \langle \omega_{+}(\xi),k\rangle -s_1\theta_{+i}(\xi)- s_2\theta_{+j}(\xi)
-2\pi w|\geqslant
\frac{2\alpha_{+}}{(2+|k|)^{\tau}},
$$
where  $\xi \in \Pi_{+}$ for all $k \in \mathbb{Z}^{n}$
$(0< |k|\leq K_{+})$, $0\leqslant |s_1|+|s_2|\leqslant 2$, and
$s_{d}\in \mathbb{Z}$  $(d=1,2)$.


Given the  choice of $T_+$,
we suppose that $h_+\leq \frac{5}{6}h$ For $\xi  \in \Pi_{h_{+}}^{+}$, it follows the Cauchy estimate that

$$|\partial(\omega_{+}(\xi)-\omega(\xi))/ \partial\xi|_{h_{+}}\leq \frac{|\omega_{+}(\xi)-\omega(\xi)|_{h}}{h-h_{+}}\leq \frac{6\epsilon}{h}.
$$
Letting $T_+=T+\frac{6\epsilon}{h}$  and $ h_+=\frac{\alpha_+}{T_+(2+K_+)^{\tau+1}}$, we obtain
$$ \max_{\xi\in \Pi_{h_{+}}}| \partial \omega_{+}/ \partial \xi|
\leqslant \max_{\xi\in \Pi_{h_{+}}}| \partial (\omega_{+}-\omega(\xi))/ \partial \xi|
+\max_{\xi\in \Pi_{h_{+}}}| \partial \omega/ \partial \xi|
\leq T_+,$$
and $$ \max_{\xi\in \Pi_{h_{+}}}| \partial \theta_{+}/ \partial \xi|
\leq T_+.$$


\subsection{Iteration}
Set
\begin{gather*}
 s_0=s,\quad \rho_0=(1-\sigma)s/10,\quad r_0=r,\quad \alpha_0=\alpha,\\
 \eta_0=E_0,\quad  \epsilon_0=\alpha_0^{2\bar{\nu}}\rho_0^{2\nu}E_0{\eta_0^{2}},
\quad \frac{e^{-K_0\rho_0}}{\eta_0^{2}}=E_0.
\end{gather*}
Let
\begin{gather*}
\omega_0(\xi)=\omega(\xi),\quad
 \theta_0(\xi)=\big(\theta_{01}(\xi),\theta_{02}(\xi),\dots, \theta_{0m}(\xi)\big)\\
\begin{aligned}
\Pi_0= \Big\{& \xi\in \Pi: |\langle \omega_0(\xi),k\rangle -s_1\theta_{0i}(\xi)
- s_2\theta_{0j}(\xi)-2\pi w|\geqslant \frac{2\alpha}{(1+|k|)^{\tau}},\\
& k\in Z^{n},\; 0<|k|\leq K_0,\; 0\leqslant |s_1|+|s_2|\leqslant 2,\;
s_{d}\in \mathbb{Z},\;d=1,2 \Big\}.
  \end{aligned}
\end{gather*}
Let
$$
T_0=T=\max_{\xi\in \Pi_{h}} \big\{| \frac{\partial \omega(\xi)}{\partial \xi} |,\,
|\frac{\partial \theta(\xi)}{\partial \xi} |\big\},\quad
h_0=\frac{\alpha_0}{(2+K_0)^{\tau+1}T_0}.
$$
Assume that $ \rho_j,  s_j,  r_j, E_j, \alpha_j, T_j$ are
well-defined for the $j$-th step.
Then we define $\eta_j, K_j, \epsilon_j, h_j$ as follows:
\begin{gather}\label{e3.24}
\eta_j=E_j,\quad \epsilon_j=\alpha^{2\bar{\nu}}_j\rho_j^{2\nu}E_j{\eta_j^{2}},\\
\label{e3.25}
\frac{e^{-K_j\rho_j}}{\eta_j^{2}}=E_j,\quad
 h_j=\frac{\alpha_j}{(1+K)_j^{\tau+1}T_j}.
\end{gather}

Define  the inductive sequences:
\begin{gather}\label{e3.26}
\rho_{j+1}=\sigma\rho_j,\quad s_{j+1}=s_j-5\rho,\quad r_{j+1}=\eta_j r_j,\\
\label{e3.27}
\alpha_{j+1}=\alpha_j-(1+K_j)^{\tau+1}\epsilon_j,\quad
 E_{j+1}=cE_j^{\frac{4}{3}},\quad T_{j+1}=T_j+\frac{6\epsilon_j}{d_j}.
\end{gather}
Let
\begin{align*}
\Pi_{j+1}= \Big\{& \xi\in \Pi_j:
 |\langle \omega_{j+1}(\xi),k\rangle -s_1\theta_{j+1i}(\xi)- s_2\theta_{j+1z}(\xi)
 -2\pi w| \geqslant \frac{2\alpha_{j+1}}{(|k|+2)^{\tau}},\\
& K_j<|k|\leq K_{j+1},\; 0\leqslant |s_1|+|s_2|\leqslant 2,\;
 s_{d}\in \mathbb{Z,}~d=1,2 \Big\}
\end{align*}
and
$$
\Pi_{{j+1}_{h_{j+1}}}=\Big\{ \xi\in C^{n}: \operatorname{dist}( \xi,
\Pi_{j+1})\leqslant h_{j+1} \Big \}.
$$

 The proofs of the following two Lemmas are similar to the idea described
in \cite{po,xu07}.  To make the paper self-contained, we present our proofs
in the Appendix.

\begin{lemma}\label{10.101}
In view of definitions of parameters in \eqref{e3.24}-\eqref{e3.27}, we have
\begin{gather}\label{eq3.28}
h_{j+1}\leq \frac{5}{6} h_j,\quad
\max_{ \xi\in \Pi_{h_{j+1}}}
\Big\{\big| \frac{\partial \omega_{j+1}(\xi)}{\partial \xi} \big|,
\big|\frac{\partial \theta_{j+1}(\xi)}{\partial \xi} \big|\Big\} \leq T_{j+1}, \\
 T_0\leq T_j\leq T_0+1,\quad \frac{1}{2}\alpha_j\leq\alpha_{j+1}\leq\alpha_j.
\end{gather}
\end{lemma}

\begin{remark} \rm
By the KAM iteration theory and Lemma \ref{10.101}, the KAM step can
iterate infinitely times.
\end{remark}

We now provide some useful estimates on the  Gevrey-smoothness and
convergence of the iteration.
Let
\begin{equation}\label{e3.28}
D_j^{\beta}=\frac{c\alpha_{j-1}^{\bar{\nu}}\rho_{j-1}^{\nu}
E^{3}_{j-1}\beta!}{h_j^{|\beta!|}}\quad \text{and}\quad
 J_j^{\beta}=\frac{c\epsilon_{j-1}\beta!}{h_j^{|\beta!|}}.
\end{equation}
Then a straightforward calculation can lead to the following result.

\begin{lemma}\label{10.102}
If $D_j^{\beta}$ and $J_j^{\beta}$ are defined by \eqref{e3.28}, then
\begin{gather*}
D_j^{\beta}\leq c \rho_j^{\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},\\
J_j^{\beta}\leq c  \rho_j^{2\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},
\end{gather*}
where $M= \frac{2T+1}{\alpha}[ \frac{4(\mu-1)(n+1)}{3}]^{\mu-1},\mu=\tau+\delta$
and $ c$ only depends on $n, \alpha$ and $\mu$.
\end{lemma}

Using the generating functions $\langle p, q_+ \rangle + F_j(p,q_+)$ to define
$\{\Psi_j(\cdot;\xi)\}$,
 the Cauchy estimate  gives
$$
\|\Psi_j-id\|_{r_j;D(s_j-3\rho_j,r_j)\times\Pi_{h_j}}\le
\frac{c \epsilon_j }{\alpha_j^{\bar{\nu}} \rho_j^{\nu}}, \ \
\|D\Psi_j-Id\|_{r_j;D(s_j-3\rho_j,r_j)\times\Pi_{h_j}}\le
\frac{c \epsilon_j }{\alpha_j^{\bar{\nu}} \rho_j^{\nu+1}}.
$$
Let  $\Psi^{j}=\Psi_1\circ\Psi_2\circ \dots  \circ\Psi_j$.
Then we have
$\{\Phi_{j+1}(\cdot;\xi)=(\Psi^{j})^{-1}\circ \Phi_j\circ\Psi^{j}\}$,
generated by $H_{j+1}(\cdot;\xi)=N_{j+1}+P_{j+1}$,
where
$$
N_{j+1}=\langle x+\omega_{j+1}(\xi),\hat{y}\rangle+\langle A_{
 j+1}u,\hat{v}\rangle+\frac{1}{2}\langle B_{
 j+1}u,u\rangle+\frac{1}{2}\langle C_{
 j +1}\hat{v},\hat{v}\rangle$$
with
\begin{gather*}
|\omega_{j+1}-\omega_j|\le \epsilon_j,\quad
|\theta_{j+1}-\theta_j|\le c\epsilon_j,\quad \forall j \geqslant 1,\\
\|X_{P_{j+1}}\|_{ r_{j+1}; \mathcal{D}(s_{j+1},  r_{j+1})
\times\Pi_{h_{j+1}}}\le \epsilon_{j+1}.
\end{gather*}

\subsection{Convergence of the KAM iteration}
 Following  \cite{xu07,zhang061,zhang062}, we have
\begin{gather*}
\|\Psi^{j}-\Psi^{j-1}\|_{r_j;D(s_j-3\rho_j,r_j)\times\Pi_{h_j}}
\leq c \alpha_{j-1}^{\bar{\nu}}\rho_{j-1}^{\nu} E^{3}_{j-1}, \\
\| D(\Psi^{j}-\Psi^{j-1})\|_{r_j;D(s_j-3\rho_j,r_j)\times\Pi_{h_j}}
\leq \alpha_{j-1}^{\bar{\nu}}\rho_{j-1}^{\nu+1} E^{3}_{j-1}.
\end{gather*}
By the Cauchy  estimate and Lemma \ref{10.102}, we have
\begin{gather*}
 \| \partial^{\beta}_{\xi}(\Psi^{j}-\Psi^{j-1})\|_{r_j;D(s_j-3\rho_j,r_j)
\times\Pi_j}
\leq \rho_j^{\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},\\
 \| \partial^{\beta}_{\xi}D(\Psi^{j}-\Psi^{j-1})\|_{r_j;D(s_j-3\rho_j,r_j)
 \times\Pi_j}
\leq   \rho_j^{\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},\\
 \| \partial^{\beta}_{\xi}(\omega_j-\omega_{j-1})\|_{\Pi_j}
\leq \rho_j^{2\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},\\
\| \partial^{\beta}_{\xi}(\theta_j-\theta_{j-1})\|_{\Pi_j}
\leq \rho_j^{2\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}}.
\end{gather*}

Since $ s_j\to s/2$, $r_j\to 0$, and $h_j\to 0$ as $j\to\infty$, we define
$$
D_{*}=D(\frac{s}{2},0),\quad
\Pi_{*}=\cap_{j\geq 0}\Pi_j\quad  \text{and} \quad
\Psi_{*}=\lim_{j\to \infty}\Psi^{j}.
$$
So we have  $\partial^{\beta}_{\xi}\Psi^{j} \to \partial^{\beta}_{\xi}\Psi^{*}$
on $ D(\frac{s}{2},\frac{r}{2})$ and
$$
\|\partial^{\beta}_{\xi}(\Psi_{*}-id)\|_{\frac{r}{2};D(\frac{s}{2},\frac{r}{2})
\times\Pi_{*}}\leq c\rho_0^{\nu} M^{|\beta|}\beta!^{\mu}E_0^{\frac{9}{4(n+1)}}
$$
for $\beta\in Z^{+}_{n}$. Thus, we arrive at \eqref{542}.

Let  $\omega_{*}=\lim_{j\to \infty}\omega_j$ and
$\theta_{*}=\lim_{j\to \infty}\theta_j$
We then have
\begin{gather*}
| \partial^{\beta}_{\xi}(\omega_{*}(\xi)-\omega(\xi))|_{\Pi_{*}}
\leq c\rho_0^{2\nu}M^{|\beta|}\beta!^{\mu}E_0^{\frac{9}{4(n+1)}}, \\
 | \partial^{\beta}_{\xi}(\theta_{*}(\xi)-\theta(\xi))|_{\Pi_{*}}
\leq c\rho_0^{2\nu}M^{|\beta|}\beta!^{\mu}E_0^{\frac{9}{4(n+1)}}
\end{gather*}
for all $\beta\in Z^{+}_{n}$.
Thus, we arrive at the desired result \eqref{543} and \eqref{9.118}
Moreover, we have
$$
  |\langle \omega_{*}(\xi),k\rangle -s_1\theta_{*i}(\xi)
- s_2\theta_{*j}(\xi)-2\pi w|\geq \frac{\alpha_{*}}{(2+|k|)^{\tau}},
$$
where $\xi \in \Pi_{*}$, $0 \neq k\in Z^{n}$,
$0\leqslant |s_1|+|s_2|\leqslant 2$, $s_{d}\in \mathbb{Z}$
$(d=1,2)$, $\alpha_{*}=\lim_{j\to \infty}\alpha_j$ with
$\frac{\alpha_0}{2}<\alpha_{*}<\alpha_0$
This implies that \eqref{9.119} holds.


\subsection{Estimate of measure }
We consider the measure of the subset
$\Pi_*$ such that the small divisor conditions
\eqref{s0}-\eqref{s2} hold for all $\omega_j, \theta_j,\alpha_j$
 and $j\geq 1$

Recalling  \eqref{531}, we know that the frequency $ \omega_j(\xi)$
satisfies \eqref{531}.  Thus, we can follow the same approach as
in \cite{xu94,xu07} to obtain the estimate for $\Pi_*$.
   Here we omit the details.

\section{Appendix}

\begin{proof}[Proof of Lemma \ref{10.101}]
From the definitions of $E_j$ and $\rho_j$, we have
$ E_j\leq (cE_0)^{(4/3)^{j}}$
Letting $x_j=K_j\rho_j=-\ln E_j^{3}$, we have
$$
\frac{K_{j+1}}{K_j}=\frac{1}{2}\frac{\ln c}{\ln E_j}+\frac{4}{3\sigma}.
$$
Let $E_0$ be small enough such that
$$
-\frac{\ln c}{\ln E_j}\leq(1-\sigma)\frac{4}{3}.
$$
Then we get
$$
\frac{4}{3}\leq \frac{K_{j+1}}{K_j}\leq\frac{4}{3 \rho}.
$$
Moreover, for a sufficiently small $E_0$, we have that $ 24<K_j<K_{j+1}$
Then we have
$$
\frac{h_{j+1}}{h_j}=\frac{\alpha_{j+1}}{\alpha_j} \cdot
\frac{T_j}{T_{j+1}} \cdot \frac{(2+K_j)^{\tau}}{(2+K_{j+1})^{\tau}}\leq\frac{5}{6}.
$$
Clearly, $h_{j+1}\leq \frac{5}{6} h_j$ and so the assumption
$h_{+}\leq \frac{5}{6} h$ holds.
Suppose that
$$
\max_{ \xi\in \Pi_{h_j}} \Big\{\big| \frac{\partial \omega_j(\xi)}{\partial \xi}\big|,
\big|\frac{\partial \theta_j(\xi)}{\partial \xi}\big|\} \leq T_j.
$$
From \eqref{e3.27}, we know that $T_{j+1}=T_j+\frac{6\epsilon_j}{d_j}$.
Since $h_{j+1}\leq \frac{5}{6} h_j$ and $|\omega_{j+1}-\omega_j|\leqslant \epsilon$,
we have
\begin{align*}
|\frac{\partial \omega_{j+1}}{\partial\xi}|
&=|\frac{\partial (\omega_{j+1}- \omega_j +\omega_j)}{\partial\xi}  |\\
&\leq |\frac{\partial (\omega_{j+1}- \omega_j )}{\partial\xi}|
 +|\frac{\partial \omega_j}{\partial\xi}|
\leq T_{j+1}
\end{align*}
and similarly,
$$
|\frac{\partial \theta_{j+1}}{\partial\xi} | \leqslant T_{j+1}.
$$
Consequently, by  mathematical induction we obtain the desired result
 \eqref{eq3.28}.


From the definitions of $T_j, h_j$ and $\epsilon_j$, we have
\begin{align*}
T_{j+1}
 &= T_j+\frac{6\epsilon_j}{d_j} \\
 &= T_0+\sum_{i=0}^{j}\frac{6\epsilon_{i}}{h_{i}} \\
 &= T_0+6 \sum_{i=0}^{j} (x_{i})^{2\nu}e^{-x_{i}}T_{i}.
\end{align*}
Let $E_0$ be sufficiently small such that
$$
 \sum_{i=0}^{j} (x_{i})^{2\nu}e^{-x_{i}}T_{i} \leq \frac{1}{6},
$$
then we have $ T_0\leq T_j\leq T_0+1$.

Note that $ \alpha^{2\bar{\nu}}_j\leqslant \alpha_j$ and
$(2+K_j)^{\tau+1}\leqslant (3K_j)^{2\nu}$. Then we have
\begin{gather*}
 \alpha_j^{2\bar{\nu}}\rho_j^{2\nu}(2+K_j)^{2\nu}E^{3}_j
\leqslant \alpha_j(3\rho_jK_j)^{2\nu}E^{3}_j
\leqslant \alpha_j(3x_j)^{2\nu}e^{-x_j},\\
 \alpha_{j+1}=\alpha_j-(2+K_j)^{\tau+1}\epsilon_j
\geqslant \alpha_j\left(1-(3x_j)^{2\nu}e^{-x_j}\right).
\end{gather*}
If  $ E_0$ is sufficiently small, then it gives
$$
\prod_{j=1}^{\infty}(1-(3x_j)^{2\nu}e^{-x_j})=1-O(x_0^{-1})>\frac{1}{2}.
$$
Thus, we obtain
$$
\frac{1}{2}\alpha_j\leq\alpha_{j+1}\leq\alpha_j.
$$
\end{proof}


\begin{proof}[Proof of Lemma \ref{10.102}]
By the choices of parameters, we have
\begin{align*}
\rho_{j+1} x_{j+1}^{\frac{\delta}{\tau+1}}
&= \rho_{j+1} K_{j+1}^{\frac{\delta}{\tau+1}}\rho_{j+1}^{\frac{\delta}{\tau+1}}\\
&\geqslant  \Big(\frac{4}{3} \Big)^{\frac{\delta}{\tau+1}}
\sigma^{\frac{\delta+\tau+1}{\tau+1}}   \rho_j   \rho_j^{\frac{\delta}{\tau+1}}
K_j^{\frac{\delta}{\tau+1}}\\
&= \Big(\frac{4}{3} \Big)^{\frac{\delta}{\tau+1}}
 \sigma^{\frac{\delta+\tau+1}{\tau+1}}   \rho_j x_j^{\frac{\delta}{\tau+1}}.
\end{align*}

Choosing $\sigma=(\frac{3}{4})^{\frac{\delta}{\delta+\sigma+1}}$, we get
$\left(\frac{4}{3}\right)^{\frac{\delta}{\tau+1}}
\sigma^{\frac{\delta+\tau+1}{\tau+1}} \geqslant 1$
Since  $ \rho_0 x_0^{\frac{\delta}{\tau+1}}\geq1$, we have
$  \rho_j x_j^{\frac{\delta}{\tau+1}}\geq1$ for all $j\geqslant1$,
and hence $ \frac{1}{\rho_j}\leq x_j^{\frac{\delta}{\tau+1}}$
So we have
$$
K_j=\frac{x_j}{\rho_j} \leq x_j^{1+\frac{\delta}{\tau+1}},
$$
which implies that
$ K_j^{\tau+1}\leq x_j^{\tau+1+\delta}$
In view of $ h_j= \frac{\alpha_j}{(K+2)_j^{\tau+1}T_j}$,
$T_j<T+1$, $\frac{1}{2}\alpha\leq\alpha_j$
and $E_{j-1}= E_j^{\frac{3}{4}}=e^{-\frac{x_j}{4}}$,
we have
\begin{align*}
 D_j^{\beta}
&\leq  c \alpha^{\bar{\nu}}\rho_j^{\nu} \beta!
\Big( \frac{T+1}{\frac{\alpha}{2}}\Big)^{|\beta|}
\left(x_j^{\tau+1+\delta} \right)^{|\beta|}e^{-3x_j/4} \\
&\leq  c \rho_j^{\nu}\Big( \frac{2(T+1)}{\alpha}\Big)^{|\beta|} \beta!
e^{-\frac{3x_j}{4}\frac{1}{n+1}}
 \Big[ x_j^{\beta_1}e^{-\frac{3x_j}{4}\frac{1}{(\tau+\delta)(n+1)}}
\dots x_j^{\beta_{n}}e^{-\frac{3x_j}{4}\frac{1}{(\tau+\delta)(n+1)}}
\Big]^{\tau+\delta}\\
&\leq  c \rho_j^{\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}},
\end{align*}
where $M= \frac{2T+1}{\alpha}[ \frac{4(\mu-1)(n+1)}{3}]^{\mu-1}$,
$\mu=\tau+\delta$ and $ c$ only depends on $ n,\alpha$ and $\mu$.

In an analogous manner, we can derive
$$
J_j^{\beta}\leq c  \rho_j^{2\nu}M^{|\beta|}\beta!^{\mu}E_j^{\frac{9}{4(n+1)}}.
$$
\end{proof}


\subsection*{Acknowledgements}
The  author would like to thank Professor  Mark Levi  for his
hospitality and valuable discussions, and thank the Department of Mathematics
of Pennsylvania State University for generous support during his visiting
from February 7, 2016-February 8, 2017.
This work was supported by NSF of Jiangsu Higher Education Institutions
of China under No.  14KJB110009, and  by the NSF of Jiangsu Province
under No. BK20140927 and No. BK20150934.

\begin{thebibliography}{99}

\bibitem{ar63} V. Arnold;
\emph{Proof of  A.N. Kolmogorov's  theorem on conservation of conditionally
periodic motions under small perturbations of the hamiltonian function},
Uspeki Matematicheskii Nauk, (18) 1963, 13--40.

\bibitem{br} A. D. Bruno;
\emph{Analytic form of differential equations},
Trudy Moskovskogo Matematicheskogo Obshchestva,  26 (1972), 199-239.

\bibitem{bi} Q.-Y. Bi,  J.-X. Xun;
\emph{Persistence of Lower Dimensional Hyperbolic Invariant Tori for Nearly
Integrable Symplectic Mappings},  Qualitative Theory of Dynamical Systems,
13 (2014),  269-288.

\bibitem{ccq} C.-Q. Cheng,  Y.-S. Sun;
\emph{Existence of invariant tori in three-dimensional measure-preserving mappings},
Celestial Mechanics and Dynamical Astronomy, 47 (1989), 275-292.

\bibitem{De}  R. de la Llave,  J. M. James;
\emph{Parameterization of invariant manifolds by reducibility for
volume preserving and symplectic maps},  Discrete Contin. Dynam. Systems, 32 (2012),
 4321-4360.

\bibitem{ko} A. N. Kolmogorov;
\emph{On preservation of conditionally periodic motions under a small change
 in the Hamiltonian function},  Dokl. Akad. Nauk SSSR,  98 (1954), 527-530.


\bibitem{lu} X.-Z. Lu, J. Li,  J.-X. Xu;
\emph{A KAM Theorem for a Class of Nearly Integrable Symplectic Mappings},
Journal of Dynamics and Differential Equations, (2015), 1-24.

\bibitem{DM} H. R. Dullin, J. D. Meiss;
\emph{Resonances and twist in volume-preserving mappings},
SIAM Journal on Applied Dynamical Systems,  11 (2012), 319-349.

\bibitem{FDM} A. M. Fox, J. D. Meiss;
\emph{Greene's residue criterion for the breakup of invariant tori of
volume-preserving maps}, Physica D: Nonlinear Phenomena, 243(2013), 45-63.

\bibitem{G} V. Gelfreich,  C. Sim\'o, A. Vieiro;
\emph{ Dynamics of 4D symplectic maps near a double resonance},
Physica D: Nonlinear Phenomena, 243 (2013), 92-110.

\bibitem{g} S. M. Graff;
\emph{On the conservation of hyperbolic invariant tori for Hamiltonian systems},
Journal of Differential Equations,  15 (1974), 1-69.

\bibitem{me65}  V. K. Melnikov;
\emph{On some cases of conservation of conditionally periodic motions under
a small change of the Hamiltonian function},  Soviet Math. Doklady, 6 (1965),
1592-1596.

\bibitem{me68}  V. K. Melnikov;
\emph{A family of conditionally periodic solutions of a Hamiltonian system},
Soviet Math. Doklady,  9(1968), 822-886.

\bibitem{m} J. Moser;
\emph{On invariant curves of area-preserving mappings of an annulus},
Vandenhoeck \& Ruprecht, 1962.

\bibitem{mo} J.  Moser;
\emph{Convergent series expansions for quasi-periodic motions},
Mathematische Annalen, 169 (1967), 136-176.

\bibitem{pon1899} H. Poincar\'e, R. Magini;
\emph{Les m\'ethodes nouvelles de la mécanique c\'eleste},
Il Nuovo Cimento (1895-1900), 1 0(1899),  128-130.

\bibitem{pon1912} H. Poincar\'e;
\emph{Sur un th\'eoreme de g\'eom\'etrie},
Rendiconti del Circolo Matematico di Palermo (1884-1940),  33(1912), 375-407.

\bibitem{p5}  J. P\"oschel;
\emph{Integrability of Hamiltonian systems on Cantor sets},
  Communications on Pure and Applied Mathematics,  35(1982), 653-696.

\bibitem{p2}  J. P\"oschel;
\emph{On elliptic lower dimensional tori in Hamiltonian systems},
Mathematische Zeitschrift,  202(1989), 559-608.

\bibitem{po} G. Popov;
\emph{Invariant Tori, Effective Stability, and Quasimodes with Exponentially
Small Error Terms I Birkhoff Normal Forms},  Annales Henri Poincar\'e,
 Birkh\"auser Verlag,  1(2000), 223-248.

\bibitem{po1} G. Popov;
\emph{KAM theorem for Gevrey Hamiltonians},
Ergodic Theory and Dynamical Systems, 24 (2004), 1753-1786.

\bibitem{r1}  H. R\"ussmann;
\emph{On the existence of invariant curves of twist mappings of an annulus},
Geometric dynamics,  Springer Berlin Heidelberg, 677-718 (1983).

\bibitem{r2} H. R\"ussmann;
\emph{On twist Hamiltonian}, Talk on the Colloque
International, M\'ecanique c\'eleste et systemes hamiltoniens,
Marseille, 1990.

\bibitem{sh}  Z.-J. Shang;
\emph{A note on the KAM theorem for symplectic mappings},
Journal of Dynamics and Differential Equations,  12(2000), 357-383.

\bibitem{wang} S.-P. Wang,  D.-F. Zhang,  J.-X. Xu;
\emph{On the persistence of elliptic lower-dimensional tori in Hamiltonian
systems under the first Melnikov condition and R\"ussmann's non-degeneracy
condition},  Nonlinear Analysis: Theory, Methods  Applications, 66(2007),
 1675-1685.

\bibitem{w} H.  Whitney;
\emph{Analytic extensions of differentiable functions defined in closed sets},
Transactions of the American Mathematical Society, 36 (1934), 63-89.

\bibitem{xia} Z.-H. Xia;
\emph{Existence of invariant tori in volume-preserving diffeomorphisms},
Ergodic Theory and Dynamical Systems,  12 (1992), 621-631.

\bibitem{xu94} J.-X. Xu, J.-G. You, Q.-J.Qiu;
\emph{Invariant tori for nearly integrable Hamiltonian systems with degeneracy},
 Mathematische Zeitschrift,  226 (1997), 375-387.

\bibitem{xu07} J.-X. Xu, J.-G. You;
\emph{Gevrey-smoothness of invariant tori for analytic nearly integrable
Hamiltonian systems under R\"ussmann's non-degeneracy condition},
Journal of Differential Equations,  235 (2007), 609-622.

\bibitem{zhang061} D.-F.  Zhang,  J.-X. Xu;
\emph{On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems
under R\"ussmann's non-degeneracy condition},
Discrete Contin. Dyn. Syst. Ser. A, 16 (2006), 635-655.

\bibitem{zhang062} D.-F. Zhang,  J.-X. Xu;
\emph{Gevrey-smoothness of elliptic lower-dimensional invariant tori
in Hamiltonian systems under R\"ussmanns non-degeneracy condition},
J. Math. Anal. Appl. 322(2006), 293-312

\bibitem{zhu} W. Zhu, B. Liu,  Z. Liu;
\emph{The hyperbolic invariant tori of symplectic mappings}.
Nonlinear Analysis: Theory, Methods  Applications,  68(2008), 109-126.

\end{thebibliography}

\end{document}