# Exponential Stability Estimate of Symplectic Integrators for Integrable
Hamiltonian Systems

Zhaodong Ding111School of Mathematical Sciences, Inner Mongolia University,
Hohhot 010021, China; e-mail<EMAIL_ADDRESS>Zaijiu
Shang222Institute of Mathematics, Academy of Mathematics and System Sciences,
Chinese Academy of Sciences, Beijing 100080, P. R. China; e-mail:
<EMAIL_ADDRESS>Bo Xie333South China University of Technology, Guangzhou,
P. R. China; e-mail<EMAIL_ADDRESS>
###### Abstract

We prove a Nekhoroshev-type theorem for nearly integrable symplectic map. As
an application of the theorem, we obtain the exponential stability symplectic
algorithms. Meanwhile, we can get the bounds for the perturbation, the
variation of the action variables, and the exponential time respectively.
These results provide a new insight into the nonlinear stability analysis of
symplectic algorithms. Combined with our previous results on the numerical KAM
theorem for symplectic algorithms (2018), we give a more complete
characterization on the complex nonlinear dynamical behavior of symplectic
algorithms.

##  1 Introduction

After the pioneering work of Channel (1983), Feng Kang (1985, 1986) and Ruth
(1983), the symplectic integrator has become a widely interested subject on
the problem of numerically solving Hamiltonian systems. Extensive computer
experimentation, by some typical models of Hamiltonian systems, has shown the
overwhelming superiority of symplectic algorithms over the conventional non-
symplectic ones, especially in simulating the global and structural dynamic
behavior of the systems (e.g. see [14] and [11]). The symplectic algorithm,
which applied to integrable Hamiltonian system, may be characterized as a
perturbation of the phase flow of the integrable system. Here the smallness of
the perturbation is described by the time-step size of the algorithm which
also enters into the frequency map of the integrable system. Therefore
numerical stability problem arises.

We consider a nearly integrable symplectic map
$\mathcal{C}:(I,\theta)\rightarrow(\hat{I},\hat{\theta})$ generated by an
analytical function in action-angle variables of the form
$H(\hat{I},\theta)=H_{0}(\hat{I})+h(\hat{I},\theta)$ which is defined on
$\mathcal{G}\times T^{n}$, where actions space $\mathcal{G}$ is an open and
bounded domain of $\mathbb{R}^{n}$ and $h$ is a small perturbation, of size
$\epsilon$. The symplectic map $\mathcal{C}$ is given implicitly by

$\displaystyle\hat{I}$ $\displaystyle=I-\partial_{2}h(\hat{I},\theta)~{},$
$\displaystyle\hat{\theta}$ $\displaystyle=\theta+\partial
H_{0}(\hat{I})+\partial_{1}h(\hat{I},\theta)~{}.$

It is clear that the symplectic map is integrable when the perturbation $h$
vanishes. In this case the dynamics of the symplectic map is trivial. The
action variables remain constant for all iterative times and the angle
variables vary linearly with respect to the iteration steps. However, a
perturbed symplectic map may generate very complicated dynamics in general
when the perturbation $h$ is non-vanishing. It is believed that there exists
unstable motions. For example, Arnold diffusion may take place if the degrees
of freedom are higher than two in mapping case [2].

By KAM theorem, nevertheless, most of motions are perpetually stable for
sufficiently smooth nearly integrable Hamiltonian systems, and these stable
ones form a Cantor set with large measure in the phase space [1, 21]. On the
other hand, all motions are exponentially stable if the systems are analytic
and the unperturbed integrable part of the Hamiltonian satisfies the so-called
steepness condition by Nekhoroshev theorem. The result was first proved by
Nekhoroshev for nearly integrable Hamiltonian systems and are also called
effective stability in literature. More precisely, he proved for sufficiently
small $\epsilon$ and for all initial values $I(0)$ in action space, one has

$|I(t)-I(0)|_{2}\leq C\epsilon^{b}\qquad\mbox{for}\quad|t|\leq
T_{0}\exp(\epsilon^{-a})~{},$

with constants $C,~{}T_{0},~{}a$, and $b$, provided the steepness conditions
are fulfilled for $H_{0}$. Further improvements on the stability exponents $a$
and $b$ were made by Pöschel [19], Lochak [16, 17] and Bounemoura [5].

In his paper, Nekhoroshev conjectured similar result will hold for the nearly
integrable symplectic map. Later, Kuksin and Pöschel [15] gave a proof about
the exponential stability of nearly integrable symplectic maps by proving the
existence of a non-autonomous analytic Hamiltonian system interpolating a
symplectic map and applying the Nekhoroshev theorem of the Hamiltonian case.
However, the proof is an existence one and there is not explicit estimate
about the small perturbation. In 2004, Guzzo [12] proposed a direct proof of
the Nekhoroshev theorem for nearly integrable symplectic maps. This result is
very important and valuable, but his estimate for the perturbation is not
suitable to be applied to the small twist problem and thus the symplectic
integrator. Here we get a different estimate of the perturbation so that it
can be applied to the small twist maps and symplectic integrators. In [23],
Nekhoroshev stability of symplectic algorithms has been considered. But there
the Nekhoroshev stability can be obtained only when the order of the algorithm
is greater than 2. In this article we completely solve this problem.

In the present paper, we obtain an exponential stability result by
construction, and provide explicit estimates of the involved quantities for
the nearly integrable symplectic map, then apply it to the cases of small
twist maps and symplectic integrators. Following the original idea of
Nekhoroshev [18], the proof of the main theorem is divided into three steps.
First, normal forms of the nearly integrable symplectic map are constructed on
some subdomains of phase space that are known as resonant blocks. Second,
these normal forms lead to stability estimates on the corresponding
subdomains. The third step is to give a geometric construction which
guarantees the whole action space can be completely covered by such resonant
blocks.

###  1.1 Notations

We introduce some notations used in this paper, most of which are from [9].
Given $\rho=(\rho_{1},\rho_{2})\geq 0$ (i.e. $\rho_{j}\geq 0$, $j=1,2$), we
first introduce the sets

$\mathcal{V}_{\rho_{1}}(G):=\\{I\in\mathbb{C}^{n}~{}|~{}|I-I^{\prime}|_{2}\leq\rho_{1}~{}\mbox{for
some}~{}I^{\prime}\in G\\},$

and

$\mathcal{W}_{\rho_{2}}(T^{n}):=\\{\theta\in\mathbb{C}^{n}/(2\pi\mathbb{Z}^{n})~{}|~{}\texttt{Re}\theta\in
T^{n},~{}|\texttt{Im}\,\theta|_{\infty}\leq\rho_{2}\\},$

where $|\cdot|_{2}$ and $|\cdot|_{\infty}$ denote, respectively, the Euclidean
norm and the maximum norm for vectors; $\texttt{Re}\theta$ and
$\texttt{Im}\,\theta$ denote the real part and the imaginary part of $\theta$
respectively. Then define

$\mathscr{D}_{\rho}(G):=\mathcal{V}_{\rho_{1}}(G)\times\mathcal{W}_{\rho_{2}}(T^{n})\,.$

Several kinds of norms are used along this paper. First, we consider functions
of the $n$ action variables. Given a (real or complex) function $f(I)$,
defined on a complex neighborhood $\mathcal{V}_{\eta}(G)$, we introduce the
supremum norm

$|f|_{G,\,\eta}:=\sup_{I\in\mathcal{V}_{\eta}(G)}|f(I)|\,,\qquad|f|_{G}:=|f|_{G,\,0}\,.$

In this way, the subscript $\eta$ is removed from the notation if $\eta=0$.
This remark applies throughout this section.

In an analogous way, we consider the supremum norm for vector-valued
functions, i.e., vectorfields. Given
$F:\mathcal{V}_{\eta}(G)\rightarrow\mathbb{C}^{n}$ and $1\leq p\leq\infty$, we
define

$|F|_{G,\,\eta,\,p}:=\sup_{I\in\mathcal{V}_{\eta}(G)}|F(I)|_{p}\,,\qquad|F|_{G,\,\eta}:=|F|_{G,\,\eta,\,2}\,.$

In this definition, $|\cdot|_{p}$ means the p-norm for vectors in
$\mathbb{C}^{n}$, i.e. $|v|_{p}=(\sum_{j=1}^{n}|v_{j}|^{p})^{1/p}$ for $1\leq
p<\infty$, and $|v|_{\infty}=\max_{1\leq j\leq n}|v_{j}|$ .

Next we consider functions of the action-angle variables. For a given complex
function $f(I,\theta)$ ($2\pi$-periodic in $\theta$) defined on the
neighborhood $\mathscr{D}_{\rho}(G),~{}\rho=(\rho_{1},\,\rho_{2})\geq 0$, we
may consider its supremum norm

$|f|_{G,\,\rho}:=\sup_{(I,\,\theta)\in\mathscr{D}_{\rho}(G)}|f(I,\theta)|\,.$

But if $f$ is analytic on (a neighborhood of ) the set
$\mathscr{D}_{\rho}(G)$, we may define an exponentially weighted norm in terms
of the Fourier series of $f$. Writing
$f(I,\theta)=\sum_{k\in\mathbb{Z}^{n}}f_{k}(I)e^{ik\cdot\theta}$, we introduce

$||f||_{G,\,\rho}:=\sum_{k\in\mathbb{Z}^{n}}|f_{k}|_{G,\,\rho_{1}}\cdot
e^{|k|_{1}\rho_{2}}\,.$

Note that $|f|_{G,\,\rho}\leq||f||_{G,\,\rho}$ .

Exactly in the same way as before we may extend the definitions of the norms
to the case of vector-valued functions. Given
$F:\mathscr{D}_{\rho}(G)\rightarrow\mathbb{C}^{n}$ and $1\leq p\leq\infty$,
and writing $F(I,\theta)=\sum_{k\in\mathbb{Z}^{n}}F_{k}(I)e^{ik\cdot\theta}$,
where $F_{k}:\mathcal{V}_{\rho_{1}}(G)\rightarrow\mathbb{C}^{n}$, we define

$||F||_{G,\,\rho,\,p}:=\sum_{k\in\mathbb{Z}^{n}}|F_{k}|_{G,\,\rho_{1},\,p}\cdot
e^{|k|_{1}\rho_{2}}\,,\qquad||F||_{G,\,\rho}:=||F||_{G,\,\rho,\,2}\,.$

The Cauchy estimates about the Fourier norms are provided by [19]. That is, if
$f$ analytic on $\mathscr{D}_{\rho}(G)$, for $0<\delta<\rho$ one has

$\Big{|}\Big{|}\frac{\partial f}{\partial
I}\Big{|}\Big{|}_{G,\,(\rho_{1}-\delta_{1},\,\rho_{2}),\,\infty}\leq\frac{1}{\delta_{1}}||f||_{G,\,\rho}\,,\qquad\Big{|}\Big{|}\frac{\partial
f}{\partial\theta}\Big{|}\Big{|}_{G,\,(\rho_{1},\,\rho_{2}-\delta_{2}),\,1}\leq\frac{1}{e\delta_{2}}||f||_{G,\,\rho}\,.$

Finally, for $Df=(\partial f/\partial I,\,\partial f/\partial\theta)$ we
introduce the vectorfield norm

$||Df||_{G,\,\rho,\,c}:=\max\Big{(}\Big{|}\Big{|}\frac{\partial
f}{\partial\theta}\Big{|}\Big{|}_{G,\,\rho,\,1},\,c\Big{|}\Big{|}\frac{\partial
f}{\partial I}\Big{|}\Big{|}_{G,\,\rho,\,\infty}\Big{)},$

where $c>0$ is a parameter to be fixed in subsequent sections.

###  1.2 Main Result

The main theorem for nearly integrable symplectic map can be stated as
follows.

Theorem 1. Let $H(\hat{I},\theta)=H_{0}(\hat{I})+h(\hat{I},\theta)$ be
analytic in $\mathscr{D}_{\sigma}(\mathcal{G})$, where $\mathcal{G}$ is an
open bounded domain of $\mathbb{R}^{n}$, and $\sigma=(\sigma_{1},\sigma_{2})$
is positive. Let $\omega(\hat{I})=\partial H_{0}(\hat{I})$ satisfying

$m|I_{1}-I_{2}|_{2}\leq|\omega(I_{1})-\omega(I_{2})|_{2}\leq
M|I_{1}-I_{2}|_{2}$ (1)

for $I_{1},I_{2}\in\mathcal{V}_{\sigma_{1}}(\mathcal{G})$ with positive
constants $m,\,M$.

Assume

$\epsilon=||h||_{\mathcal{G},\,\sigma}\leq\min(\epsilon_{0},\,\sigma_{2}^{b})\,,$

where

$\epsilon_{0}=\frac{M^{2}\cdot\sigma_{1}^{4}}{\pi^{2}(21n+30)^{2}\big{[}(\frac{2M}{m})^{n}\cdot
n!\big{]}^{4}}\,,$

$b=2(n^{2}+n+2)$ and $\sigma_{1}<\frac{1}{4M}$ .

Then the symplectic map $\mathcal{C}$ which generated by
$H(I^{\prime},\theta)$ satisfies

$|I(t)-I_{0}|_{2}\leq\triangle\quad\mbox{for}\quad|t|\leq
T\quad\mbox{and}\quad I_{0}\in\mathcal{G}-\triangle\,,$

where $(I(t),\theta(t))=\mathcal{C}^{t}(I_{0},\theta_{0})$ ($t$ is viewed as
iterative times),

$\displaystyle\triangle$ $\displaystyle=c_{0}\epsilon^{\frac{1}{b}}\quad$ with
$\displaystyle c_{0}=\frac{8nM}{3m}(3n+2)\sigma_{1}\,,$ $\displaystyle T$
$\displaystyle=T_{0}\epsilon^{-\frac{3}{4}}e^{c_{1}\epsilon^{-\frac{1}{b}}}\quad$
with $\displaystyle
T_{0}=\frac{(\frac{m}{2M})^{n}\sigma_{1}\sigma_{2}}{2^{7}n!}\quad\mbox{and}\quad
c_{1}=\frac{\sigma_{2}}{24}\,.$

The rest of this paper is devoted to the proof of the main theorem. In Section
2, the analytic part is presented which concerns the construction of the
normal form with exponentially small remainder on resonant blocks. The
geometric part, in Section 3, concerns the covering of the whole action space
$\mathcal{G}$ by a family of resonant blocks. In Section 4, the proof of
Theorem 1 is finished through making choices for the free parameters. Finally,
as applications of the main theorem, we consider the stability of the small
twist type symplectic map and symplectic integrator which applied to the
integrable Hamilton system in Section 5.

##  2 The Analytic Part

At first, we transform the mapping $\mathcal{C}$ by the partial coordinates
stretching $\mathcal{W}_{\gamma}:(x,y)\rightarrow(I,\theta)=(\gamma x,y)$ ,
and obtain a new mapping
$T_{\gamma}=\mathcal{W}_{\gamma}^{-1}\circ\mathcal{C}\circ\mathcal{W}_{\gamma}:(x,y)\rightarrow(\hat{x},\hat{y})$
to be defined in the new phase space $\mathcal{G}_{\gamma}\times T^{n}$ by

$\left\\{\begin{aligned} \hat{x}&=x-\partial_{2}F(\hat{x},y)\\\
\hat{y}&=y+\partial_{1}F(\hat{x},y)\end{aligned}\right.$

where

$F(x,y)=F_{0}(x)+f(x,y)$

is well defined on $\mathcal{G}_{\gamma}\times T^{n}$ with

$F_{0}=\gamma^{-1}H_{0}(\gamma x),\quad f(x,y)=\gamma^{-1}h(\gamma x,y)$

and

$\mathcal{G}_{\gamma}=\\{x\in\mathbb{R}^{n}~{}|~{}\gamma
x\in\mathcal{G}\\}\,.$

For the time being, $\gamma$ is considered as a free parameter. $F(x,y)$ is
real analytic in $\mathcal{D}_{\widetilde{\sigma}}(\mathcal{G}_{\gamma})$,
where $\widetilde{\sigma}=(\widetilde{\sigma}_{1},\widetilde{\sigma}_{2})$
with $\widetilde{\sigma}_{1}=\gamma^{-1}\sigma_{1}$ and
$\widetilde{\sigma}_{2}=\sigma_{2}$. Accordingly, the frequency map of the
integrable mapping associated to the generating function $F_{0}$ turns into
$\widetilde{\omega}(x)=\partial F_{0}(x)$ and the condition satisfied by the
map $\widetilde{\omega}$ turns out to be

$\gamma
m|x_{1}-x_{2}|_{2}\leq|\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2})|_{2}\leq\gamma
M|x_{1}-x_{2}|_{2}$ (2)

for $x_{1},x_{2}\in\mathcal{V}_{\widetilde{\sigma}_{1}}(\mathcal{G}_{\gamma})$
. In addition, we have

$\tilde{\epsilon}:=||f||_{\mathcal{G}_{\gamma},\,\widetilde{\sigma}}=\gamma^{-1}||h||_{\mathcal{G},\,\sigma}=\gamma^{-1}\epsilon\,.$

From now on, we fix $\gamma=\beta M^{-1}$, where $\beta>0$ is a parameter to
be determined later. Denoting $\mu=\frac{m}{M}$ , we have

$\beta\mu|x_{1}-x_{2}|_{2}\leq|\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2})|_{2}\leq\beta|x_{1}-x_{2}|_{2}$
(3)

In order to transform the generating function $F(x,y)=F_{0}(x)+f(x,y)$ of
$T_{\gamma}$ to the normal form, we need seek for a suitable canonical
transformation $\Phi$, that is constructed iteratively as a product of the
successive approximately identical canonical transformation
$\Phi^{(1)},~{}\Phi^{(2)},\ldots$. In doing that, one meets the small
denominators $1-e^{ik\cdot\widetilde{\omega}(x)}$, which in general vanish in
a dense subset of $\mathcal{G}_{\gamma}$ . These resonances are given by the
equations $k\cdot\widetilde{\omega}(x)+2\pi l=0$, for
$k\in\mathbb{Z}^{n},~{}l\in\mathbb{Z}$. As usual, the small denominators
$1-e^{ik\cdot\widetilde{\omega}(x)}$ should been excluded in the process of
constructing the normal form. However, it is not necessary to take care of all
the small denominators. We can do that in a certain subdomain which can
guarantee $|1-e^{ik\cdot\widetilde{\omega}(x)}|\geq\alpha$ for $|k|\leq K$ and
$k\notin\mathcal{M}$, where $\alpha$ and $K$ are independent parameters and
$\mathcal{M}$ is a given sublattice of $\mathbb{Z}^{n}$.

Next we give the related notations in detail. Let $\mathcal{M}$ be a
sublattice of $\mathbb{Z}^{n}$. We only consider the maximal ones, that is not
properly contained in any other sublattice of the same dimension. A maximal
sublattice $\mathcal{M}$ with dim$\mathcal{M}=r$, is said $K$-lattice if it
admits a basis $(k^{(1)},\ldots,k^{(r)})$ satisfying $|k^{(j)}|_{1}\leq K$ for
$j\leq r$ holds, and such a basis will be called a $K$-basis ([3]). A function
$g(x,y)$ is said to be in a normal form with respect to $\mathcal{M}$ of
degree $K$ if its Fourier series expansion in the angular variables is
restricted to the form $g(x,y)=\sum_{k\in\mathcal{M},\,|k|_{1}\leq
K}g_{k}(x)e^{ik\cdot y}$. We express this by writing
$g\in\mathcal{R}(\mathcal{M},K)$. Note that a function is in the normal form
with respect to the trivial modulo $\mathcal{M}=0$ if it does not depend on
the angular variables.

We restrict ourselves to a subset $G\subset\mathcal{G}_{\gamma}$, where the
frequency vectors $\widetilde{\omega}(x)$ is allowed to satisfy some resonance
relations corresponding to a fixed sublattice $\mathcal{M}$, but a
neighborhood of all other resonances of order less than or equal to $K$ are
excluded. In precise, a subset $G\subset\mathcal{G}_{\gamma}$ is said to be
$\alpha,K$-nonresonant modulo $\mathcal{M}$ if

$|1-e^{ik\cdot\widetilde{\omega}(x)}|\geq\alpha\quad\mbox{for
all}~{}\,k\in\mathbb{Z}^{n}_{K}\backslash\mathcal{M}~{}\,\mbox{and}~{}\,x\in
G\,,$

where $\mathbb{Z}^{n}_{K}:=\\{k\in\mathbb{Z}^{n}~{}|~{}|k|_{1}\leq K\\}$.

A special situation arises when $\mathcal{M}$ is the trivial sublattice
$\Theta$ of $\mathbb{Z}^{n}$ containing only 0. In this case the set $G$ is
said to be completely $\alpha,K$-nonresonant. In the corresponding normal
form, $g$ is independent of the angle variables. Naturally, the analysis of
this case is simpler than in the presence of resonances.

The nonresonance condition on the set $G$ can be extended to a complex
neighborhood of small enough radius $\rho_{1}$.

Lemma 2.1 Let $F_{0}(x)$ be a real analytic function in
$\mathcal{V}_{\rho_{1}}(G)$, and let $\widetilde{\omega}=\partial F_{0}$.
Assume that $G$ is $\alpha,K$-nonresonant modulo $\mathcal{M}$ $(0<\alpha\leq
1)$, and $\widetilde{\omega}$ satisfies (3). If

$\rho_{1}\leq\frac{\alpha}{4K\beta}\,,$

then $\mathcal{V}_{\rho_{1}}(G)$ is $\alpha/2,K$-nonresonant modulo
$\mathcal{M}$.

Proof. $\forall x\in\mathcal{V}_{\rho_{1}}(G)$, there exists $\tilde{x}\in G$
such that $|x-\tilde{x}|_{2}\leq\rho_{1}$ by definition. Because

$\begin{split}\big{|}1-e^{ik\cdot\widetilde{\omega}(x)}\big{|}&\geq\big{|}1-e^{ik\cdot\widetilde{\omega}(\tilde{x})}\big{|}-\big{|}e^{ik\cdot\widetilde{\omega}(x)}-e^{ik\cdot\widetilde{\omega}(\tilde{x})}\big{|}\\\
&\geq\alpha-\big{|}e^{ik\cdot\widetilde{\omega}(x_{*})}<k,\,\widetilde{\omega}(\tilde{x})-\widetilde{\omega}(x)>\big{|}\,,\end{split}$

where for the last inequality we have used the mean value theorem with
$x_{*}\in\mathcal{V}_{\rho_{1}}(G)$. Thus, there exists $x^{\prime}\in G$,
s.t. $|x_{*}-x^{\prime}|_{2}\leq\rho_{1}$ and
$|\bar{x}_{*}-\bar{x}^{\prime}|_{2}\leq\rho_{1}$, where $\bar{x}$ is the
complex conjugation of $x$. Noting that $x^{\prime}=\bar{x}^{\prime}$, we have
$|x_{*}-\bar{x}_{*}|_{2}\leq 2\rho_{1}$. Therefore,

$\big{|}e^{ik\cdot\widetilde{\omega}(x_{*})}\big{|}\leq
e^{|k|_{1}\cdot|\texttt{Im}\,\widetilde{\omega}(x_{*})|_{2}}\leq
e^{K\cdot\frac{|\widetilde{\omega}(x_{*})-\widetilde{\omega}(\bar{x}_{*})|_{2}}{2}}\leq
e^{K\beta\rho_{1}}\leq 2$

and
$\big{|}e^{ik\cdot\widetilde{\omega}(x_{*})}<k,\,\widetilde{\omega}(\tilde{x})-\widetilde{\omega}(x)>\big{|}\leq
2K\beta\rho_{1}\leq\frac{\alpha}{2}$ . That completes our proof. ∎

Next we state the Normal Form Lemma as follows.

Lemma 2.2 (Normal Form Lemma) Let $\mathcal{M}\subseteq\mathbb{Z}^{n}$ be a
$K$-lattice, $0<\alpha\leq 1$ and $\beta\leq\alpha$. $F(x,y)=F_{0}(x)+f(x,y)$
is analytic in $\mathcal{D}_{\rho}(G)$, $\rho=(\rho_{1},\rho_{2})$. Suppose
that $\mathcal{V}_{\rho_{1}}(G)$ is $\alpha,K$-nonresonant modulo
$\mathcal{M}$ and the frequency map $\widetilde{\omega}$ satisfies (3). If

$||Df||_{G,\,\rho,\,c}\leq\frac{\alpha\,\rho_{1}}{\widetilde{C}AK\rho_{2}}\,,$
(4)

where $\widetilde{C}=21n+30,~{}A=1+\frac{\beta c}{\alpha}e^{K\beta\rho_{1}}$
and $\rho_{1}\leq\min(\frac{\alpha}{4K\beta},1)$, then there exists a real
analytic canonical transformation
$\Phi:\mathcal{D}_{\frac{\rho}{2}}(G)\rightarrow\mathcal{D}_{\rho}(G)$ such
that the conjugate symplectic map $T_{\gamma}^{\prime}=\Phi^{-1}\circ
T_{\gamma}\circ\Phi:\mathcal{D}_{\frac{\rho}{2}}(G)\rightarrow\mathcal{D}_{\rho}(G)$
is generated by the analytic function $F^{\prime}=F_{0}+Z+R$ with
$Z\in\mathcal{R}(\mathcal{M},K)$. Moreover, the following hold

1) $||DZ||_{G,\,\frac{\rho}{2},\,c}\leq 2||Df||_{G,\,\rho,\,c}$ .

2) $||DR||_{G,\,\frac{\rho}{2},\,c}\leq
3e^{-\frac{K\rho_{2}}{12}}||Df||_{G,\,\rho,\,c}$ .

3) $|P_{x}\Phi-id|_{G,\,\frac{\rho}{2}}\leq\frac{\rho_{1}}{2^{8}}$ ,
where $P_{x}$ denotes the projection onto $x$-coordinates.

In order to prove the Normal Form Lemma, we need some technique lemmas.

Lemma 2.3 [9] Let $f$ be an analytic function in $\mathcal{D}_{\rho}(G)$. For
$0<\delta=(\delta_{1},\delta_{2})<\rho$ and $c>0$ given, let us denote

$\hat{\delta}_{c}:=\min(\delta_{1},c\delta_{2})\,.$

Then,

(a)
$||Df||_{G,\,\rho-\delta,\,c}\leq\frac{c}{\hat{\delta}_{c}}||f||_{G,\,\rho}\,.$

(b) $||D(f^{>K})||_{G,\,\rho-\delta,\,c}\leq
e^{-K\delta_{2}}||Df||_{G,\,\rho,\,c}\,,$ where
$f^{>K}(x,y)=\sum\limits_{|k|_{1}>K}f_{k}(x)e^{ik\cdot y}$ .

The following lemma is similar with Lemma 6 in [12], and the way of the proof
can be found there.
Lemma 2.4 [12] Let $T_{\gamma}$ and $\Phi$ be symplectic maps which generated
by analytic functions $F$ and $\chi$ respectively. $T_{\gamma},\Phi:G\times
T^{n}\rightarrow G\times T^{n}$ defined by

$\displaystyle\Phi:(a,\varphi)\rightarrow(x,y)$
$\displaystyle\left\\{\begin{aligned} x&=a+\partial_{2}\chi(a,y)\\\
y&=\varphi-\partial_{1}\chi(a,y)\end{aligned}\right.$ (5) $\displaystyle
T_{\gamma}:(x,y)\rightarrow(\hat{x},\hat{y})$
$\displaystyle\left\\{\begin{aligned} \hat{x}&=x-\partial_{2}F(\hat{x},y)\\\
\hat{y}&=y+\partial_{1}F(\hat{x},y)\end{aligned}\right.$ (6)

Then one of the generating functions of the conjugate symplectic map
$T_{\gamma}^{\prime}=\Phi^{-1}\circ
T_{\gamma}\circ\Phi:(a,\varphi)\rightarrow(\hat{a},\hat{\varphi})$ is

$\widetilde{F}(\hat{a},\varphi)=a\cdot\varphi-a\cdot
y+\hat{a}\cdot\hat{y}-\hat{a}\cdot\varphi+\hat{x}\cdot
y-\hat{x}\cdot\hat{y}+F(\hat{x},y)+\chi(\hat{a},\hat{y})-\chi(a,y)\,,$

where variables $x,\hat{x},y,\hat{y},a,\hat{\varphi}$ are functions of the
independent variables $\hat{a},\varphi$ .

Lemma 2.5 Let $\chi(x,y)$ be analytic in $\mathcal{D}_{\rho}(G)$, and given
positive numbers $\delta=(\delta_{1},\delta_{2})<(\rho_{1},\rho_{2})$. Assume
$||D\chi||_{G,\,\rho,\,c}\leq\frac{\hat{\delta}_{c}}{2}$, then the symplectic
transformation $\Phi$ generated by the function $\chi$ is well defined in
$\mathcal{D}_{\rho-\delta}(G)$. Furthermore, one has

$\Phi(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho-\frac{\delta}{2}}(G)\quad\mbox{and}\quad\Phi^{-1}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho-\frac{\delta}{2}}(G).$

Proof. Let
$\mathcal{A}_{\delta_{1},\,\delta_{2}}=\\{v:\mathcal{D}_{\rho-\delta}(G)\rightarrow\mathbb{C}^{n}~{}|~{}||v||_{G,\,\rho-\delta}\leq\frac{\delta_{2}}{2}\\}$.
It is easy to know that $\mathcal{A}_{\delta_{1},\,\delta_{2}}$ is a closed
bounded subset of a Banach space. Now consider the map
$\mathcal{F}(v)(x,y)=\frac{\partial\chi(x,y-v(x,y))}{\partial x}$ , which is
well defined for any $v\in\mathcal{A}_{\delta_{1},\,\delta_{2}}$ and maps the
space into itself from the fact that

$||\mathcal{F}||_{G,\,\rho-\delta}\leq\frac{1}{c}||D\chi||_{G,\,\rho,\,c}\leq\frac{\hat{\delta}_{c}}{2c}\leq\frac{\delta_{2}}{2}\,.$
(7)

Moreover, for any $v_{1},v_{2}\in\mathcal{A}_{\delta_{1},\,\delta_{2}}$, one
has

$\displaystyle||\mathcal{F}(v_{1})(x,y)-\mathcal{F}(v_{2})(x,y)||_{G,\,\rho-\delta}$
$\displaystyle=\Big{|}\Big{|}\frac{\partial\chi(x,y-v_{1}(x,y))}{\partial
x}-\frac{\partial\chi(x,y-v_{2}(x,y))}{\partial
x}\Big{|}\Big{|}_{G,\,\rho-\delta}$
$\displaystyle=\Big{|}\Big{|}\frac{\partial^{2}\chi(x,y^{*})}{\partial
x\partial y}\cdot(v_{1}-v_{2})\Big{|}\Big{|}_{G,\,\rho-\delta}$
$\displaystyle\leq\frac{1}{\delta_{1}}\Big{|}\Big{|}\frac{\partial\chi}{\partial
y}\Big{|}\Big{|}_{G,\,\rho,1}\cdot||v_{1}-v_{2}||_{G,\,\rho-\delta}$
$\displaystyle\leq\frac{1}{2}||v_{1}-v_{2}||_{G,\,\rho-\delta}\,.$

This shows that the map
$\mathcal{F}:\mathcal{A}_{\delta_{1},\,\delta_{2}}\rightarrow\mathcal{A}_{\delta_{1},\,\delta_{2}}$
is contractive. Therefore, there exists a unique
$v^{*}\in\mathcal{A}_{\delta_{1},\,\delta_{2}}$ such that
$\mathcal{F}(v^{*})=v^{*}$ and the symplectic transformation
$\Phi:(x,y)\rightarrow(\hat{x},\hat{y})$ can be expressed explicitly in the
form

$\left\\{\begin{aligned} \hat{x}&=x+\frac{\partial\chi(x,y-v^{*})}{\partial
y}\\\ \hat{y}&=y-\frac{\partial\chi(x,y-v^{*})}{\partial
x}\end{aligned}\right.$

which is well defined and real analytic for any
$(x,y)\in\mathcal{D}_{\rho-\delta}(G)$. It is easy to show
$\Phi(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho-\frac{\delta}{2}}(G)$
by means of (7) and

$\Big{|}\Big{|}\frac{\partial\chi(x,y-v^{*})}{\partial
y}\Big{|}\Big{|}_{G,\,\rho-\delta,\,1}\leq||D\chi||_{G,\,\rho,\,c}\leq\frac{\delta_{1}}{2}\,.$

In similar way, we can prove that
$\Phi^{-1}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho-\frac{\delta}{2}}(G)$
. Specifically, denote the set
$\widetilde{\mathcal{A}}_{\delta_{1},\,\delta_{2}}=\\{u:\mathcal{D}_{\rho-\delta}(G)\rightarrow\mathbb{C}^{n}~{}|~{}||u||_{G,\,\rho-\delta}\leq\frac{\delta_{2}}{2}\\}$
and consider the map
$\widetilde{\mathcal{F}}(u)(x,y)=\frac{\partial\chi(x+u(x,y),y)}{\partial y}$
, which is well defined in $\widetilde{\mathcal{A}}_{\delta_{1},\,\delta_{2}}$
. ∎

Lemma 2.6 Let $F(\hat{x},y)=F_{0}(\hat{x})+f(\hat{x},y)$ be analytic in
$\mathcal{D}_{\rho}(G)$ with $\widetilde{\omega}(x)=\partial F_{0}(x)$
satisfying (3). Given positive numbers
$\delta=(\delta_{1},\delta_{2})<(\rho_{1},\rho_{2})$, if

$\rho_{1}\leq\frac{\delta_{2}}{6\beta}\quad\mbox{and}\quad||Df||_{G,\,\rho,\,c}\leq\frac{\hat{\delta}_{c}}{2}\,,$

then the symplectic map $T_{\gamma}$ generated by $F$ is well defined in
$\mathcal{D}_{\rho-\delta}(G)$ and

$T_{\gamma}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{\delta_{2}}{3}}(G)\,,\quad\quad
T_{\gamma}^{-1}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{\delta_{2}}{3}}(G).$

Proof. Similar to the proof of Lemma 2.5, we know that for any
$(x,y)\in\mathcal{D}_{\rho-\delta}(G)$, there exists a unique
$\hat{x}\in\mathcal{V}_{\rho_{1}-\frac{\delta_{1}}{2}}(G)$ such that
$\hat{x}=x-\frac{\partial f(\hat{x},y)}{\partial y}$. Therefore,
$\hat{y}=y+\widetilde{\omega}(\hat{x})+\frac{\partial
f(\hat{x},y)}{\partial\hat{x}}$ is also well defined in
$\mathcal{D}_{\rho-\delta}(G)$. Moreover, when
$(x,y)\in\mathcal{D}_{\rho-\delta}(G)$,

$\displaystyle|\texttt{Im}\,(\hat{y}-y)|_{\infty}$
$\displaystyle\leq|\texttt{Im}\,\widetilde{\omega}(\hat{x})|_{G,\,\rho_{1}-\delta_{1}/2,\,\infty}+\Big{|}\Big{|}\frac{\partial
f}{\partial x}\Big{|}\Big{|}_{G,\,\rho,\,\infty}$
$\displaystyle\leq\Big{|}\frac{\widetilde{\omega}(\hat{x})-\widetilde{\omega}(\bar{\hat{x}})}{2}\Big{|}_{G,\,\rho_{1}-\delta_{1}/2,\,\infty}+\frac{1}{c}||Df||_{G,\,\rho,\,c}$
$\displaystyle\leq\frac{\beta}{2}|\hat{x}-\bar{\hat{x}}|_{2}+\frac{\delta_{2}}{2}\,.$

Due to $\hat{x}\in\mathcal{V}_{\rho_{1}-\frac{\delta_{1}}{2}}(G)$, there
exists $x^{\prime}\in G$ such that $|\hat{x}-x^{\prime}|_{2}\leq\rho_{1}$ .
Thus

$|\hat{x}-\bar{\hat{x}}|_{2}\leq|\hat{x}-x^{\prime}|_{2}+|\bar{x^{{}^{\prime}}}-\bar{\hat{x}}|_{2}\leq
2\rho_{1}\leq\frac{\delta_{2}}{3\beta}\,.$

Therefore, $|\texttt{Im}\,(\hat{y}-y)|_{\infty}\leq\frac{2}{3}\delta_{2}\,,$
and we get

$T_{\gamma}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{\delta_{2}}{3}}(G)\,.$

In similar way, one can prove that
$T_{\gamma}^{-1}(\mathcal{D}_{\rho-\delta}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{\delta_{2}}{3}}(G)$
.∎

Combing Lemma 2.4, 2.5 and 2.6, we have

Lemma 2.7 Consider the generating function
$F(\hat{x},y)=F_{0}(\hat{x})+f(\hat{x},y)$ and $\chi$, which are analytic in
$\mathcal{D}_{\rho}(G)$. let $\widetilde{\omega}(x)=\partial F_{0}(x)$
satisfies (3). Given positive numbers $\delta=(\delta_{1},\delta_{2})$ such
that $3\delta\leq\rho$ . Suppose

$\rho_{1}\leq\frac{\delta_{2}}{6\beta}\,,\quad||Df||_{G,\,\rho,\,c}\leq\frac{\hat{\delta}_{c}}{2}\quad\mbox{and}\quad||D\chi||_{G,\,\rho,\,c}\leq\frac{\hat{\delta}_{c}}{2}\,.$

Let $T_{\gamma}$ be the symplectic map generated by $F$, $\Phi$ by $\chi$ and
$T_{\gamma}^{\prime}=\Phi^{-1}\circ T_{\gamma}\circ\Phi$ . Then
$T_{\gamma}^{\prime}$ and ${T_{\gamma}^{\prime}}^{-1}$ are analytic and
symplectic diffemorphisms which defined in
$\mathcal{D}_{\rho_{1}-2\delta_{1},\,\rho_{2}-3\delta_{2}}(G)$. It is

$T_{\gamma}^{\prime}(\mathcal{D}_{\rho_{1}-2\delta_{1},\,\rho_{2}-3\delta_{2}}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{4}{3}\delta_{2}}(G)\,,$

and

${T_{\gamma}^{\prime}}^{-1}(\mathcal{D}_{\rho_{1}-2\delta_{1},\,\rho_{2}-3\delta_{2}}(G))\subseteq\mathcal{D}_{\rho_{1}-\frac{\delta_{1}}{2},\,\rho_{2}-\frac{4}{3}\delta_{2}}(G)\,.$

In addition, the conjugate symplectic map $T_{\gamma}^{\prime}$ can be
generated by a function $\widetilde{F}$ which is analytic in a domain
containing $\mathcal{D}_{\rho-3\delta}(G)\,.$

Now, we state and prove an iterative lemma, which is then used in the proof of
Normal Form Lemma.

Lemma 2.8 (Iterative Lemma) Consider
$F(\hat{x},y)=F_{0}(\hat{x})+Z(\hat{x},y)+R(\hat{x},y)$ real analytic in
$\mathcal{D}_{\rho}(G)$ and $\mathcal{V}_{\rho_{1}}(G)$ is
$\alpha,K$-nonresonance modulo $\mathcal{M}$,
$Z\in\mathcal{R}(\mathcal{M},K)$. Let the symplectic map
$T_{\gamma}:(x,y)\rightarrow(\hat{x},\hat{y})$ be given by the generating
function $F(\hat{x},y)$, and let $\widetilde{\omega}(x)=\partial F_{0}(x)$
satisfying the condition (3) where $\beta\leq\alpha$. Assume

$||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c}\leq\frac{\alpha\hat{\delta}_{c}}{4A}\quad\mbox{with}\quad
3\delta\leq\rho\,,~{}\rho_{1}\leq\min\Big{(}\frac{\delta_{2}}{6\beta},\,1\Big{)}$
(8)

where $A=1+\frac{\beta c}{\alpha}e^{K\beta\rho_{1}}$ . Then, there exists a
real analytic canonical transformation $\Phi:(a,\varphi)\rightarrow(x,y)$ such
that the conjugate symplectic map $T_{\gamma}^{\prime}=\Phi^{-1}\circ
T_{\gamma}\circ\Phi$ is generated by
$\widetilde{F}(\hat{a},\varphi)=F_{0}(\hat{a})+\widetilde{Z}(\hat{a},\varphi)+\widetilde{R}(\hat{a},\varphi)$
with $\widetilde{Z}\in\mathcal{R}(\mathcal{M},K)$ and $\widetilde{F}$ is
analytic in a domain containing $\mathcal{D}_{\rho-3\delta}(G)$. In addition,
ones have

(a)

$||D\widetilde{Z}||_{G,\,\rho-3\delta,\,c}\leq||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c}\,.$

(b)

$||D\widetilde{R}||_{G,\,\rho-3\delta,\,c}\leq\Big{\\{}\frac{2}{\delta_{1}}\big{(}1+\frac{A}{\alpha}\big{)}\big{[}\triangle_{1}(||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c})+e^{-K\delta_{2}}\big{]}\cdot||DR||_{G,\,\rho,\,c}+\triangle_{2}(||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c})+e^{-K\delta_{2}}+\frac{2\beta
AC}{\alpha}\Big{\\}}\cdot||DR||_{G,\,\rho,\,c}$ , where

$\displaystyle\triangle_{1}$
$\displaystyle=\frac{2A}{\alpha\delta_{1}}+\frac{2}{\delta_{1}}+\frac{A}{e\alpha\delta_{2}}\Big{(}\frac{\beta
A}{\alpha}+\beta+\frac{2}{c}\Big{)}\,,$ $\displaystyle\triangle_{2}$
$\displaystyle=\frac{n+2}{\delta_{1}}\cdot\frac{A}{\alpha}+\frac{n+1}{\delta_{1}}+\frac{nA}{e\alpha\delta_{2}}\Big{(}\frac{\beta
A}{\alpha}+\beta+\frac{1}{c}\Big{)}\,.$

(c)

$|P_{x}\Phi-
id|_{G,\,\rho-3\delta}\leq\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\,.$

Proof. We define the transformation $\Phi:(a,\varphi)\rightarrow(x,y)$
implicitly with the help of a undetermined generating function $\chi$ by

$x=a+\partial_{2}\chi(a,y)\,,\qquad y=\varphi-\partial_{1}\chi(a,y)\,.$

Because of Lemma 2.7, we have the generating function
$\widetilde{F}(\hat{a},\varphi)$ of the conjugate symplectic map
$T_{\gamma}^{\prime}$ and $\widetilde{F}$ is analytic in domain containing
$\mathcal{D}_{\rho-3\delta}(G)$.

We choose $\chi$ satisfying the linear functional equation:

$R^{\leq
K}+\chi(a,y+\widetilde{\omega}(a))-\chi(a,y)=g(a,\varphi)\qquad\mbox{for
some}\quad g\in\mathcal{R}(\mathcal{M},K)\,,$

where $R^{\leq K}$ denotes the terms restricted to $|k|_{1}\leq K$ in the
Fourier expansion of $R$. By means of the Fourier expansions of $R$, $\chi$
and $g$, ones have the solutions:

$\chi_{k}(a)=\begin{cases}\frac{R_{k}(a)}{1-e^{ik\cdot\widetilde{\omega}(a)}}&~{}k\notin\mathcal{M},~{}|k|_{1}\leq
K.\\\ 0&~{}\text{else.}\end{cases}$

and

$g_{k}(a)=\begin{cases}R_{k}(a)&~{}k\in\mathcal{M},~{}|k|_{1}\leq K.\\\
0&~{}\text{else.}\end{cases}$

where

$\chi=\sum_{k\in\mathbb{Z}^{n}}\chi_{k}(a)e^{ik\cdot y}\,,\quad
g=\sum_{k\in\mathbb{Z}^{n}}\chi_{k}(a)e^{ik\cdot y}\,.$

Thus, $\chi(a,y)=\sum\limits_{\begin{subarray}{l}k\notin\mathcal{M}\\\
|k|_{1}\leq
K\end{subarray}}\frac{R_{k}(a)}{1-e^{ik\cdot\widetilde{\omega}(a)}}\cdot
e^{ik\cdot y}\,,$ and

$||\partial_{2}\chi||_{G,\,\rho,\,1}=\sum_{\begin{subarray}{l}k\notin\mathcal{M}\\\
|k|_{1}\leq
K\end{subarray}}\Big{|}\frac{kR_{k}(a)}{1-e^{ik\cdot\widetilde{\omega}(a)}}\Big{|}_{G,\,\rho_{1},\,1}\cdot
e^{|k|_{1}\cdot\rho_{2}}\leq\frac{1}{\alpha}||\partial_{2}R||_{G,\,\rho,\,1}\,,$

where in the last inequality we have used the nonresonance condition. Because

$\partial\chi_{k}(a)=\frac{\partial
R_{k}(a)}{1-e^{ik\cdot\widetilde{\omega}(a)}}+\frac{e^{ik\cdot\widetilde{\omega}(a)}[\partial_{2}R]_{k}\partial\widetilde{\omega}}{(1-e^{ik\cdot\widetilde{\omega}(a)})^{2}}\,,\qquad\mbox{for}~{}k\notin\mathcal{M},~{}|k|_{1}\leq
K\,,$

where we have used that $[\partial_{2}R]_{k}=iR_{k}(a)k$ (differentiating the
Fourier expansion of $R$). As before,
$|\texttt{Im}\,\widetilde{\omega}(a)|_{G,\,\rho_{1},\,\infty}\leq\beta\rho_{1}$.
Thus

$|\partial\chi_{k}(a)|_{G,\,\rho_{1},\,\infty}\leq\frac{1}{\alpha}|\partial
R_{k}(a)|_{G,\,\rho_{1},\,\infty}+\frac{\beta}{\alpha^{2}}e^{|k|_{1}\beta\rho_{1}}\cdot|[\partial_{2}R]_{k}|_{G,\,\rho_{1}}\,.$

Moreover,

$||\partial_{1}\chi||_{G,\,\rho,\,\infty}\leq\frac{1}{\alpha}||\partial_{1}R|_{G,\,\rho,\,\infty}+\frac{\beta}{\alpha^{2}}e^{K\beta\rho_{1}}\cdot||\partial_{2}R||_{G,\,\rho,\,1}\,.$

Therefore, we have

$\begin{split}||D\chi||_{G,\,\rho,\,c}&\leq\big{(}\frac{1}{\alpha}+\frac{\beta
c}{\alpha^{2}}e^{K\beta\rho_{1}}\big{)}\cdot||DR||_{G,\,\rho,\,c}\\\
&\triangleq\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\end{split}$ (9)

with $A=1+\frac{\beta c}{\alpha}e^{K\beta\rho_{1}}$.

Let $B=R_{0}^{-1}\circ\Phi^{-1}\circ
T_{\gamma}\circ\Phi:(a,\varphi)\rightarrow(\hat{a},\hat{\varphi})$ where
$R_{0}$ is a integrable rotation on $G\times T^{n}$ with frequency map
$\widetilde{\omega}$, i.e.
$R_{0}(a,\varphi)=(a,\varphi+\widetilde{\omega}(a))$, and $B$ can be expressed
implicitly as follows:

$\left\\{\begin{aligned}
\hat{a}&=a+\partial_{2}\chi(a,y)-\partial_{2}\chi(\hat{a},\hat{y})-\partial_{2}(Z+R)(\hat{x},y)\\\
\hat{\varphi}&=\varphi-\partial_{1}\chi(a,y)+\partial_{1}\chi(\hat{a},\hat{y})+\partial_{1}(Z+R)(\hat{x},y)-\widetilde{\omega}(\hat{a})+\widetilde{\omega}(\hat{x}).\end{aligned}\right.$
(10)

On the other hand, let
$\widetilde{Z}(\hat{a},\varphi)=Z(\hat{a},\varphi)+\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(\hat{a},\varphi)$,
where $\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R$ denotes the terms restricted
to $k\in\mathcal{M}$ and $|k|_{1}\leq K$ in the Fourier expansion of $R$, and
assume that the map $\Phi^{-1}\circ T_{\gamma}\circ\Phi$ has the form:

$\left\\{\begin{aligned}
\hat{a}&=a-\partial_{2}\widetilde{Z}(\hat{a},\varphi)-\partial_{2}\widetilde{R}(\hat{a},\varphi)\\\
\hat{\varphi}&=\varphi+\widetilde{\omega}(\hat{a})+\partial_{1}\widetilde{Z}(\hat{a},\varphi)+\partial_{1}\widetilde{R}(\hat{a},\varphi)\,.\end{aligned}\right.$

Then, $B$ has the form:

$\left\\{\begin{aligned}
\hat{a}&=a-\partial_{2}\widetilde{Z}(\hat{a},\varphi)-\partial_{2}\widetilde{R}(\hat{a},\varphi)\\\
\hat{\varphi}&=\varphi+\partial_{1}\widetilde{Z}(\hat{a},\varphi)+\partial_{1}\widetilde{R}(\hat{a},\varphi)\,.\end{aligned}\right.$
(11)

Combining (10) and (11), we have

$\displaystyle||\partial_{2}\widetilde{R}(\hat{a},\varphi)||_{G,\,\rho-3\delta,\,1}$
$\displaystyle\leq||\hat{a}-a+\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}$
$\displaystyle=||\partial_{2}\chi(a,y)-\partial_{2}\chi(\hat{a},\hat{y})-\partial_{2}(Z+R)(\hat{x},y)+\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}$
$\displaystyle\leq I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}\,,$
$\displaystyle||\partial_{1}\widetilde{R}(\hat{a},\varphi)||_{G,\,\rho-3\delta,\,\infty}$
$\displaystyle\leq||\hat{\varphi}-\varphi-\partial_{1}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle=||\partial_{1}\chi(\hat{a},\hat{y})-\partial_{1}\chi(a,y)+\partial_{1}(Z+R)(\hat{x},y)-\widetilde{\omega}(\hat{a})+\widetilde{\omega}(\hat{x})$
$\displaystyle\quad-\partial_{1}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle\leq J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}+J_{7}\,.$

where

$\displaystyle I_{1}$
$\displaystyle=||\partial_{2}\chi(\hat{a},\hat{y})-\partial_{2}\chi(a,\hat{y})||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle I_{2}$
$\displaystyle=||\partial_{2}\chi(a,\hat{y})-\partial_{2}\chi(a,y+\widetilde{\omega}(a))||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle I_{3}$
$\displaystyle=||\partial_{2}\chi(a,y+\widetilde{\omega}(a))-\partial_{2}\chi(a,y)+\partial_{2}(R(a,y)-\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(a,y))||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle I_{4}$
$\displaystyle=||\partial_{2}R(\hat{x},y)-\partial_{2}R(a,y)||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle I_{5}$
$\displaystyle=||\partial_{2}Z(\hat{x},y)-\partial_{2}Z(\hat{a},y)||_{G,\,\rho-\delta,\,1}+||\partial_{2}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(a,y)-\partial_{2}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(\hat{a},y)||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle I_{6}$
$\displaystyle=||\partial_{2}Z(\hat{a},y)-\partial_{2}Z(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}+||\partial_{2}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(\hat{a},y)-\partial_{2}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}\,,$
$\displaystyle J_{1}$
$\displaystyle=||\partial_{1}\chi(\hat{a},\hat{y})-\partial_{1}\chi(a,\hat{y})||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{2}$
$\displaystyle=||\partial_{1}\chi(a,\hat{y})-\partial_{1}\chi(a,y+\widetilde{\omega}(a))||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{3}$
$\displaystyle=||\partial_{1}\chi(a,y+\widetilde{\omega}(a))-\partial_{1}\chi(a,y)+\partial\widetilde{\omega}(a)\partial_{2}\chi(a,y+\widetilde{\omega}(a))$
$\displaystyle\quad+\partial_{1}(R(a,y)-\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}R(a,y))||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{4}$
$\displaystyle=||\partial_{1}R(\hat{x},y)-\partial_{1}R(a,y)||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{5}$
$\displaystyle=||\partial_{1}Z(\hat{x},y)-\partial_{1}Z(\hat{a},y)||_{G,\,\rho-\delta,\,\infty}+||\partial_{1}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}(R(a,y)-R(\hat{a},y))||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{6}$
$\displaystyle=||\partial_{1}Z(\hat{a},y)-\partial_{1}Z(\hat{a},\varphi)||_{G,\,\rho-\delta,\,\infty}+||\partial_{1}\mathcal{P}_{\mathcal{M}}\mathcal{T}_{K}(R(\hat{a},y)-R(\hat{a},\varphi))||_{G,\,\rho-\delta,\,\infty}\,,$
$\displaystyle J_{7}$
$\displaystyle=||\widetilde{\omega}(\hat{x})-\widetilde{\omega}(\hat{a})||_{G,\,\rho-\delta,\,\infty}+||\partial\widetilde{\omega}(a)\cdot\partial_{2}\chi(a,y+\widetilde{\omega}(a))||_{G,\,\rho-\delta,\,\infty}\,.$

Note that all the concerned variables are in $\mathcal{D}_{\rho-\delta}(G)$
and we are able to get the following estimates by middle value theorem and
Cauchy estimates.

$\displaystyle I_{1}$
$\displaystyle\leq\frac{1}{\delta_{1}}||\partial_{2}\chi||_{G,\,\rho,\,1}\cdot||\hat{a}-a||_{G,\,\rho-\delta,\,1}$
$\displaystyle\leq\frac{1}{\delta_{1}}||D\chi||_{G,\,\rho,\,c}\cdot(||\hat{a}-a+\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}+||\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1})$
$\displaystyle\leq\frac{1}{\delta_{1}}\cdot\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\cdot(||\hat{a}-a+\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}+||\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1})\,,$
where in the last inequality, we have used (9). $\displaystyle I_{2}$
$\displaystyle\leq\frac{1}{e\delta_{2}}||\partial_{2}\chi||_{G,\,\rho,\,1}\cdot||\hat{y}-y-\widetilde{\omega}(a)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle\leq\frac{1}{e\delta_{2}}||D\chi||_{G,\,\rho,\,c}\cdot||\hat{y}-y-\widetilde{\omega}(a)||_{G,\,\rho-\delta,\,\infty}\,.$

Note that

$\displaystyle||\hat{y}-y-\widetilde{\omega}(a)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle=||\widetilde{\omega}(\hat{x})-\widetilde{\omega}(a)+\partial_{1}(Z+R)(\hat{x},y)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle\leq\beta\cdot||\hat{x}-a||_{G,\,\rho-\delta,\,1}+||\partial_{1}(Z+R)(\hat{x},y)||_{G,\,\rho-\delta,\,\infty}\,,$

and $\hat{x}=a+\partial_{2}\chi(a,y)-\partial_{2}(Z+R)(\hat{x},y)$, so

$||\hat{x}-a||_{G,\,\rho-\delta,\,1}\leq||\partial_{2}\chi||_{G,\,\rho-\delta,\,1}+||\partial_{2}(Z+R)||_{G,\,\rho-\delta,\,1}.$

Thus,

$||\hat{y}-y-\widetilde{\omega}(a)||_{G,\,\rho-\delta,\,\infty}\leq\beta(||D\chi||_{G,\,\rho-\delta,\,c}+||D(Z+R)||_{G,\,\rho-\delta,\,c})+\frac{1}{c}||D(Z+R)||_{G,\,\rho-\delta,\,c}\,.$

Therefore,

$\displaystyle I_{2}$
$\displaystyle\leq\frac{A}{e\alpha\delta_{2}}||DR||_{G,\,\rho,\,c}\cdot\Big{[}\frac{\beta
A}{\alpha}||DR||_{G,\,\rho,\,c}+(\beta+\frac{1}{c})||D(Z+R)||_{G,\,\rho,\,c}\Big{]}\,.$
Similarly, we have $\displaystyle I_{3}$
$\displaystyle\leq||\partial_{2}R^{>K}||_{G,\,\rho-\delta,\,1}\leq||DR^{>K}||_{G,\,\rho-\delta,\,c}\leq
e^{-K\delta_{2}}||DR||_{G,\,\rho,\,c}\,,$ $\displaystyle I_{4}$
$\displaystyle\leq\frac{1}{\delta_{1}}||\partial_{2}R||_{G,\,\rho,\,1}\cdot||\hat{x}-a||_{G,\,\rho-\delta,\,1}\leq\frac{1}{\delta_{1}}||DR||_{G,\,\rho,\,c}\cdot\Big{(}\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}+||D(Z+R)||_{G,\,\rho,\,c}\Big{)},$
$\displaystyle I_{5}$
$\displaystyle\leq\frac{1}{\delta_{1}}||\partial_{2}Z||_{G,\,\rho,\,1}\cdot||\hat{x}-\hat{a}||_{G,\,\rho-\delta,\,1}+\frac{1}{\delta_{1}}||\partial_{2}R||_{G,\,\rho,\,1}\cdot||\hat{a}-a||_{G,\,\rho-\delta,\,1}$
$\displaystyle\leq\frac{1}{\delta_{1}}||DZ||_{G,\,\rho,\,c}\cdot\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}+\frac{1}{\delta_{1}}||DR||_{G,\,\rho,\,c}\cdot\big{(}||\hat{a}-a+\partial_{2}\widetilde{Z}||_{G,\,\rho-\delta,\,1}+||\partial_{2}\widetilde{Z}||_{G,\,\rho,\,1}\big{)}\,,$
$\displaystyle I_{6}$
$\displaystyle\leq\frac{1}{e\delta_{2}}\cdot\frac{A}{c\alpha}||DR||_{G,\,\rho,\,c}\cdot\big{(}||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c}\big{)}\,.$

By means of the condition (8) and above estimates, we get

$\displaystyle\frac{1}{2}||\hat{a}-a+\partial_{2}\widetilde{Z}(\hat{a},\varphi)||_{G,\,\rho-\delta,\,1}\leq$
$\displaystyle\bigg{[}\Big{(}\frac{2A}{\alpha\delta_{1}}+\frac{2}{\delta_{1}}+\frac{A}{e\alpha\delta_{2}}\big{(}\beta\frac{A}{\alpha}+\beta+\frac{2}{c}\big{)}\Big{)}\cdot\Big{(}||DZ||_{G,\,\rho,\,c}$
$\displaystyle+||DR||_{G,\,\rho,\,c}\Big{)}+e^{-K\delta_{2}}\bigg{]}\cdot||DR||_{G,\,\rho,\,c}\,.$

Thus,

$||\partial_{2}\widetilde{R}(\hat{a},\varphi)||_{G,\,\rho-3\delta,\,1}\leq
2\big{[}\triangle_{1}\cdot(||DR||_{G,\,\rho,\,c}+||DZ||_{G,\,\rho,\,c})+e^{-K\delta_{2}}\big{]}\cdot||DR||_{G,\,\rho,\,c}\,,$

where
$\triangle_{1}=\frac{2A}{\alpha\delta_{1}}+\frac{2}{\delta_{1}}+\frac{A}{e\alpha\delta_{2}}(\frac{\beta
A}{\alpha}+\beta+\frac{2}{c})\,.$

By estimating $J_{k},~{}k=1,\ldots,7$ in a similar way to the above and making
use of the previous estimates, we obtain

$\displaystyle J_{1}$
$\displaystyle\leq\frac{1}{\delta_{1}}||\partial_{1}\chi||_{G,\,\rho,\,\infty}\cdot||\hat{a}-a||_{G,\,\rho-\delta,\,1}$
$\displaystyle\leq\frac{1}{c\delta_{1}}\cdot\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\cdot\big{(}||\hat{a}-a+\partial_{2}\widetilde{Z}||_{G,\,\rho-\delta,\,1}+||\partial_{2}\widetilde{Z}||_{G,\,\rho-\delta,\,1}\big{)}\,,$
$\displaystyle J_{2}$
$\displaystyle\leq\frac{1}{e\delta_{2}}||\partial_{1}\chi||_{G,\,\rho}\cdot||\hat{y}-y-\widetilde{\omega}(a)||_{G,\,\rho-\delta,\,\infty}$
$\displaystyle\leq\frac{n}{e\delta_{2}}\cdot\frac{A}{c\alpha}||DR||_{G,\,\rho,\,c}\cdot\Big{(}\frac{\beta
A}{\alpha}||DR||_{G,\,\rho,\,c}+\big{(}\beta+\frac{1}{c}\big{)}||D(Z+R)||_{G,\,\rho,\,c}\Big{)}\,,$
$\displaystyle J_{3}$
$\displaystyle\leq||\partial_{1}R^{>K}||_{G,\,\rho-\delta,\,\infty}\leq\frac{1}{c}||DR^{>K}||_{G,\,\rho-\delta,\,c}\leq\frac{1}{c}e^{-K\delta_{2}}||DR||_{G,\,\rho,\,c}\,,$
$\displaystyle J_{4}$
$\displaystyle\leq\frac{n}{c\delta_{1}}||DR||_{G,\,\rho,\,c}\cdot\Big{(}\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}+||D(Z+R)||_{G,\,\rho,\,c}\Big{)},$
$\displaystyle J_{5}$
$\displaystyle\leq\frac{1}{c\delta_{1}}||DZ||_{G,\,\rho,\,c}\cdot\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}+\frac{1}{c\delta_{1}}||DR||_{G,\,\rho,\,c}\cdot\big{(}||\hat{a}-a+\partial_{2}\widetilde{Z}||_{G,\,\rho-\delta,\,1}+||\partial_{2}\widetilde{Z}||_{G,\,\rho,\,1}\big{)}\,,$
$\displaystyle J_{6}$
$\displaystyle\leq\frac{1}{\delta_{1}}\cdot\frac{A}{c\alpha}||DR||_{G,\,\rho,\,c}\cdot\big{(}||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c}\big{)}\,,$
$\displaystyle J_{7}$ $\displaystyle\leq
2\beta||\partial_{2}\chi||_{G,\,\rho-\delta,\,1}\leq
2\beta\cdot\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\,.$

Combining the above estimates, we get

$\displaystyle
c||\partial_{1}\widetilde{R}(\hat{a},\varphi)||_{G,\,\rho-3\delta,\,\infty}\leq$
$\displaystyle\Big{[}\frac{1}{\delta_{1}}\big{(}1+\frac{A}{\alpha}\big{)}\cdot||\hat{a}-a+\partial_{2}\widetilde{Z}||_{G,\,\rho-\delta,\,1}+\triangle_{2}\cdot\big{(}||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c}\big{)}$
$\displaystyle+e^{-K\delta_{2}}+\frac{2\beta
Ac}{\alpha}\Big{]}\cdot||DR||_{G,\,\rho,\,c}\,,$

where

$\triangle_{2}=\frac{n+2}{\delta_{1}}\cdot\frac{A}{\alpha}+\frac{n+1}{\delta_{1}}+\frac{nA}{e\alpha\delta_{2}}\Big{(}\frac{\beta
A}{\alpha}+\beta+\frac{1}{c}\Big{)}\,.$

Because $\delta_{1}\leq 1$, we have

$\displaystyle||DR||_{G,\,\rho-3\delta,\,c}$
$\displaystyle\leq\max\big{(}||\partial_{2}\widetilde{R}||_{G,\,\rho-3\delta,\,1},~{}c||\partial_{1}\widetilde{R}||_{G,\,\rho-3\delta,\,\infty}\big{)}$
$\displaystyle\leq\Big{\\{}\frac{2}{\delta_{1}}\big{(}1+\frac{A}{\alpha}\big{)}\big{[}\triangle_{1}(||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c})+e^{-K\delta_{2}}\big{]}\cdot||DR||_{G,\,\rho,\,c}$
$\displaystyle\quad+\triangle_{2}(||DZ||_{G,\,\rho,\,c}+||DR||_{G,\,\rho,\,c})+e^{-K\delta_{2}}+\frac{2\beta
Ac}{\alpha}\Big{\\}}\cdot||DR||_{G,\,\rho,\,c}\,.$

Finally,

$|P_{x}\Phi-
id|_{G,\,\rho-3\delta}\leq||\partial_{2}\chi||_{G,\,\rho-\delta,\,1}\leq||D\chi||_{G,\,\rho,\,c}\leq\frac{A}{\alpha}||DR||_{G,\,\rho,\,c}\,.$

∎

###  2.1 Proof of Normal Form Lemma

As the Hamiltonian case, we shall construct a series of symplectic
transformations $\Phi^{(i)}$, each of which reduces the norm of remainder by
factor $\frac{1}{e}$. After applying Iterative Lemma $N$ times, we can get an
exponentially small remainder by choosing $N=N(K)$ adequately.

Let $N\geq 1$ be an integer to be chosen below. Denoting
$\rho^{(i)}=\rho-3i\delta$, with $\delta=\frac{\rho}{6N}$. Obviously,
$\rho^{(i)}=\rho^{(i-1)}-3\delta$.

Next we apply Iterative Lemma $N$ times, and obtain a series of symplectic
transformations
$\Phi^{(i)}:\mathcal{D}_{\rho^{(i)}}(G)\rightarrow\mathcal{D}_{\rho^{(i-1)}}(G)$
for $1\leq i\leq N$. Let
$\Psi^{(i)}=\Phi^{(i)}\circ\Phi^{(i-1)}\circ\cdots\circ\Phi^{(1)}$,
$T_{\gamma}^{(i)}={\Psi^{(i)}}^{-1}\circ T_{\gamma}\circ\Psi^{(i)}$ and the
generating functions $F^{(i)}=F_{0}+Z^{(i)}+R^{(i)}$ of the symplectic map
$T_{\gamma}^{(i)}$ with $Z^{(i)}\in\mathcal{R}(\mathcal{M},K)$.

Now, we are going to show that if $\frac{K\rho_{2}}{6N}\geq 2$ then the claims
below are true for $1\leq i\leq N$.

(a)

$||DZ^{(i)}||_{G,\,\rho^{(i)},\,c}\leq\sum\limits_{j=0}^{i-1}||DR^{(j)}||_{G,\,\rho^{(j)},\,c}\,.$

(b)

$||DR^{(i)}||_{G,\,\rho^{(i)},\,c}\leq{\Large\frac{1}{e}}||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}\,.$

The proof is done by induction.

Setting $R^{(0)}=f$ and $Z^{(0)}=0$, and we choose the parameter
$c=\frac{\rho_{1}}{\rho_{2}}=\frac{\delta_{1}}{\delta_{2}}$ so that
$\hat{\delta}_{c}=\delta_{1}=c\delta_{2}$. Due to
$\frac{1}{K}\leq\frac{\rho_{2}}{12N}$ and $\widetilde{C}\geq 30$, we have

$||Df||_{G,\,\rho,\,c}\leq\frac{\alpha\rho_{1}}{\widetilde{C}AK\rho_{2}}\leq\frac{\alpha\rho_{1}}{12\widetilde{C}AN}=\frac{\alpha\delta_{1}}{2\widetilde{C}A}\leq\frac{\alpha\hat{\delta}_{c}}{60A}\,.$

Note that

$\rho_{1}\leq\frac{\alpha}{4K\beta}\leq\frac{\alpha\rho_{2}}{48N\beta}=\frac{\alpha\delta_{2}}{8\beta}\leq\frac{\delta_{2}}{6\beta}\,,$

Thus, Iterative Lemma can be applied with $\frac{\rho}{6N}$ instead of
$\delta$. Due to $\rho_{1}\leq\frac{\alpha}{4K\beta}$, we have

$A=1+\frac{\beta c}{\alpha}e^{K\beta\rho_{1}}\leq 1+\frac{2\beta
c}{\alpha}\leq 1+\frac{1}{2K\rho_{2}}\leq 2\,,$ (12)

and

$\frac{2\beta Ac}{\alpha}\leq\frac{1}{K\rho_{2}}\leq\frac{1}{12}\,.$ (13)

By means of (4) and (12), we get

$\displaystyle||DR^{(1)}||_{G,\,\rho^{(1)},\,c}$
$\displaystyle\leq\Big{(}\frac{1}{e\widetilde{C}}+\frac{n+2}{2\widetilde{C}}+\frac{n+1}{2\widetilde{C}}+\frac{n}{24e\widetilde{C}}+\frac{1}{e^{2}}+\frac{1}{12}\Big{)}\cdot||Df||_{G,\,\rho,\,c}$
$\displaystyle\leq\frac{1}{e}||Df||_{G,\,\rho,\,c}\,.$

The claim (a) is obviously true for $i=1$.

For $1<i\leq N$, note that

$\displaystyle||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}$
$\displaystyle\leq\frac{||DR^{(0)}||_{G,\,\rho,\,c}}{e^{i-1}}\leq||Df||_{G,\,\rho,\,c}\leq\frac{\alpha\hat{\delta}_{c}}{60A}\,,$
and $\displaystyle||DZ^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}$
$\displaystyle\leq\sum\limits_{j=0}^{i-2}||DR^{(j)}||_{G,\,\rho^{(j)},\,c}\leq
2||DR^{(0)}||_{G,\,\rho,\,c}\leq\frac{\alpha\hat{\delta}_{c}}{30A}\,.$

Thus, Iterative Lemma can be applied with $\frac{\rho}{6N}$ instead of
$\delta$. The claim (a) is easy to prove, and the claim (b) can be proved by
the following estimates:

$\displaystyle||DR^{(i)}||_{G,\,\rho^{(i)},\,c}$
$\displaystyle\leq\bigg{\\{}\frac{2}{\delta_{1}}\big{(}1+\frac{A}{\alpha}\big{)}\Big{[}\triangle_{1}\big{(}||DZ^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}+||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}\big{)}+e^{-K\delta_{2}}\Big{]}\cdot$
$\displaystyle\qquad||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}+\triangle_{2}\big{(}||DZ^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}+||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}\big{)}$
$\displaystyle\qquad+e^{-K\delta_{2}}+\frac{2\beta
AC}{\alpha}\bigg{\\}}\cdot||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}$
$\displaystyle\leq\bigg{\\{}\frac{2}{\delta_{1}}\big{(}1+\frac{A}{\alpha}\big{)}\Big{(}\triangle_{1}\cdot
2||Df||_{G,\,\rho,\,c}+e^{-\frac{K\rho_{2}}{6N}}\Big{)}\frac{||Df||_{G,\,\rho,\,c}}{e}+2\triangle_{2}||Df||_{G,\,\rho,\,c}$
$\displaystyle\qquad+e^{-\frac{K\rho_{2}}{6N}}+\frac{1}{12}\bigg{\\}}\cdot||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}$
$\displaystyle\leq\frac{1}{e}||DR^{(i-1)}||_{G,\,\rho^{(i-1)},\,c}\,.$

Now we may choose $N=\big{[}\frac{K\rho_{2}}{12}\big{]}$, the integer part of
$\frac{K\rho_{2}}{12}$. After iterating $N$ times, the exponential small
remainder is given by

$||DR||_{G,\,\frac{\rho}{2},\,c}=||DR^{(N)}||_{G,\,\frac{\rho}{2},\,c}\leq\frac{||Df||_{G,\,\rho,\,c}}{e^{N}}\leq
3e^{-\frac{K\rho_{2}}{12}}||Df||_{G,\,\rho,\,c}\,.$

The conclusion 3) is obtained from the fact that

$\displaystyle|P_{x}\Phi-id|_{G,\,\frac{\rho}{2}}$
$\displaystyle\leq\sum_{i=1}^{N}|P_{x}\Phi^{(i)}-id|_{G,\,\rho^{i}}\leq\sum_{i=1}^{N}\frac{A}{\alpha}||DR^{(i)}||_{G,\,\rho^{(i)},\,c}\leq\frac{2A}{\alpha}||Df||_{G,\,\rho,\,c}$
$\displaystyle\leq\frac{\rho_{1}}{2^{8}}\,,$

where the last inequality is a consequence of (4) and $\widetilde{C}\geq 48$.
Here we remark that if $K\rho_{2}\leq 12$, all results are obvious if we take
$\Phi$ as the identity map.

##  3 The Geometry of Resonances

In this section, we concern the covering of the whole action space
$\mathcal{G}_{\gamma}$ by a family of resonant blocks associated to different
lattice $\mathcal{M}$. For the symplectic map, the original geometric
construction in [3] requires some modifications (see also [12]). Here in
addition to $K$, our geometric construction will be characterized by $2n$
positive parameters
$0<\alpha_{0}=\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{n}<1$ and
$\delta_{1},\delta_{2},\cdots,\delta_{n}$. More precisely, for any choice of
these parameters, for each $K$-lattice $\mathcal{M}\subseteq\mathbb{Z}^{n}$
with dim$\mathcal{M}=r$, and some $l=(l_{1},\ldots,l_{r})\in\mathbb{Z}^{r}$,
we define:

i) Resonant manifold

$\mathcal{R}_{\mathcal{M}}^{l}=\big{\\{}x\in\mathcal{G}_{\gamma};~{}k^{(j)}\cdot\widetilde{\omega}(x)+2\pi
l_{j}=0,~{}j=1,\ldots,r\big{\\}}.$

where $k^{(1)},\ldots,k^{(r)}$ is a $K$-basis.

ii) Resonant zone

$\mathcal{Z}_{\mathcal{M}}^{l}=\big{\\{}x\in\mathcal{G}_{\gamma};~{}\big{|}k^{(j)}\cdot\widetilde{\omega}(x)+2\pi
l_{j}\big{|}\leq\alpha_{r},~{}j=1,\ldots,r\big{\\}}.$

where $k^{(1)},\ldots,k^{(r)}$ is a $K$-basis.

Note that, for $r=0$, $K$-lattice $\mathcal{M}$ is trivial, and
$\mathcal{Z}_{0}$ is defined as coinciding with the whole action space
$\mathcal{G}_{\gamma}$. Denoting by $\mathcal{Z}_{r}^{*}$, $1\leq r\leq n$,
the union of all resonant zones with the same dimension $r$, i.e.

$\mathcal{Z}_{r}^{*}=\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=r}\mathcal{Z}_{\mathcal{M}}^{l}\,,$

and set $\mathcal{Z}_{n+1}^{*}=\emptyset$.

iii) Resonant block

$\mathcal{B}_{\mathcal{M}}^{l}=\mathcal{Z}_{\mathcal{M}}^{l}\backslash\mathcal{Z}_{r+1}^{*}\,.$

Especially, $\mathcal{B}_{0}=\mathcal{Z}_{0}\backslash\mathcal{Z}_{1}^{*}$.
The dimension $r$ of $\mathcal{M}$ will also be called the multiplicity of the
corresponding resonant manifold, zone or block.

iv) Cylinder

First, let $\Pi_{\mathcal{M}}(x)$ be the hyperplane through $x$ parallel to
$\mathcal{M}$ with the same dimensionality, and denote its $\delta_{r}$
neighborhood by $\Pi_{\mathcal{M},\,\delta_{r}}(x)$, i.e.

$\Pi_{\mathcal{M},\,\delta_{r}}(x)=\big{\\{}\tilde{x}\in\mathbb{R}^{n};~{}\text{dist}(\tilde{x},\Pi_{\mathcal{M}}(x))\leq\delta_{r}\big{\\}}.$

Then, for $x\in\mathcal{B}_{\mathcal{M}}^{l}$, the cylinder is defined by

$\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x)=\Pi_{\mathcal{M},\,\delta_{r}}(x)\bigcap\mathcal{Z}_{\mathcal{M}}^{l}\,,$

Ļ Ϊ
$\Pi_{\mathcal{M},\,\delta_{r}}(x)\bigcap\partial\mathcal{Z}_{\mathcal{M}}$ .

v) Extended resonant block

$\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l}=\bigcup_{x\in\mathcal{B}_{\mathcal{M}}^{l}}\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x)\,.$

Remarks: 1\. In the definition of the resonant zone, we make the point that
$l$ is bounded because of $|k|_{1}\leq K$ and the boundedness of
$\widetilde{\omega}(x)$ and $\alpha$. In addition, the resonant zones with the
same lattice $\mathcal{M}$ don't intersect for different $l$.

2\. The resonant blocks $\mathcal{B}_{\mathcal{M}}^{l}$ constitute a covering
of the action space $\mathcal{G}_{\gamma}$ , that is
$\mathcal{G}_{\gamma}=\bigcup_{l}\bigcup_{\mathcal{M}}\mathcal{B}_{\mathcal{M}}^{l}\,.$

Now, we shall prove some properties of the geometry construction.

Proposition 3.1 (i) For any $x\in\mathcal{B}_{\mathcal{M}}^{l}$ with
dim$(\mathcal{M})=r\in\\{0,1,\ldots,n-1\\}$, if
$k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}$, then
$|k\cdot\widetilde{\omega}(x)+2\pi l_{0}|>\alpha_{r+1}$ for any
$l_{0}\in\mathbb{Z}$. In particular, for any $x\in\mathcal{B}_{0}$ it is
$|k\cdot\widetilde{\omega}(x)+2\pi l_{0}|>\alpha_{1}$ for any
$k\in\mathbb{Z}_{K}^{n}\backslash\\{0\\},\,l_{0}\in\mathbb{Z}$.
(ii)

$\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=r}\mathcal{B}_{\mathcal{M}}^{l}=\mathcal{Z}_{r}^{*}\backslash\mathcal{Z}_{r+1}^{*}\,.$

(iii)

$\mathcal{G}_{\gamma}\backslash\mathcal{Z}_{r+1}^{*}=\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})\leq
r}\mathcal{B}_{\mathcal{M}}^{l}\,.$

(iv) If $\mathcal{M}$ is $r$-dimensional $K$-lattice, $1\leq r\leq n$, then
for any $x\in\mathcal{B}_{\mathcal{M}}^{l}$,

$\text{diam}(\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x))\leq\frac{4}{\mu}\delta_{r}+\frac{2rK^{r-1}}{\beta\mu}\alpha_{r}\,.$

Proof. In order to prove the statement (i) by contradiction, assume that there
exist $l_{0}\in\mathbb{Z}$ and $x\in\mathcal{B}_{\mathcal{M}}^{l}$, such that
$|k\cdot\widetilde{\omega}(x)+2\pi l_{0}|\leq\alpha_{r+1}$ for any
$k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}$ . Note that
$\alpha_{r}\leq\alpha_{r+1}$, then we obtain that there exists a
$r+1$-dimensional $K$-lattice $\mathcal{M}^{\prime}$ and
$l^{\prime}=(l,l_{0})$ such that
$x\in\mathcal{Z}_{\mathcal{M}^{\prime}}^{l^{\prime}}$ . However,
$x\notin\mathcal{Z}_{r+1}^{*}$ in view of the definition of
$\mathcal{B}_{\mathcal{M}}^{l}$ . We get the contradiction.

For the properties (ii) and (iii), it is readily to obtain from the
definitions. In fact,

$\displaystyle\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=r}\mathcal{B}_{\mathcal{M}}^{l}$
$\displaystyle=\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=r}\big{(}\mathcal{Z}_{\mathcal{M}}^{l}\backslash\mathcal{Z}_{r+1}^{*}\big{)}$
$\displaystyle=\Big{(}\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=r}\mathcal{Z}_{\mathcal{M}}^{l}\Big{)}\backslash\mathcal{Z}_{r+1}^{*}$
$\displaystyle=\mathcal{Z}_{r}^{*}\backslash\mathcal{Z}_{r+1}^{*}\,,$

and

$\displaystyle\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})\leq
r}\mathcal{B}_{\mathcal{M}}^{l}$
$\displaystyle=\bigcup_{s=0}^{r}\Big{(}\bigcup_{l}\bigcup_{\text{dim}(\mathcal{M})=s}\mathcal{B}_{\mathcal{M}}^{l}\Big{)}$
$\displaystyle=\bigcup_{s=0}^{r}\Big{(}\mathcal{Z}_{s}^{*}\backslash\mathcal{Z}_{s+1}^{*}\Big{)}$
$\displaystyle=\mathcal{G}_{\gamma}\backslash\mathcal{Z}_{r+1}^{*}\,.$

Finally, before proving the statement (iv), we need a technique lemma which
refers to [3].

Lemma 3.1 [3] Let $k^{(1)},k^{(2)},\cdots,k^{(r)}$ be linearly independent
vectors of $\mathbb{Z}^{n}$ with $|k^{(i)}|_{1}\leq K$, and
$\omega\in\mathbb{R}^{n}$ be any linear combination of
$k^{(1)},k^{(2)},\cdots,k^{(r)}$ satisfying $|\omega\cdot
k^{(i)}|\leq\alpha,~{}(i=1,\ldots,r)$ then one has

$|\omega|_{2}<rK^{r-1}\alpha\,.$

Let's continue the proof of Proposition 3.1 (iv). For any
$x_{1},x_{2}\in\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x)$ with
$x\in\mathcal{B}_{\mathcal{M}}^{l}$, there exist
$x_{1}^{*},x_{2}^{*}\in\Pi_{\mathcal{M}}(x)$ such that
$|x_{1}^{*}-x_{1}|_{2}\leq\delta_{r}$ and
$|x_{2}^{*}-x_{2}|_{2}\leq\delta_{r}$ . Due to the convexity of the function
$F_{0}$, we conclude that

$\displaystyle\beta\mu|x_{1}-x_{2}|_{2}^{2}$
$\displaystyle\leq\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}-x_{2})\big{|}$
$\displaystyle\leq\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}-x_{1}^{*})\big{|}+\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{2}^{*})\big{|}$
$\displaystyle\quad+\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{2}^{*}-x_{2})\big{|}$
$\displaystyle\leq
2\beta|x_{1}-x_{2}|_{2}\cdot\delta_{r}+\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{2}^{*})\big{|}\,.$
(14)

Note that $x_{1}^{*}-x_{2}^{*}$ parallel to $\mathcal{M}$, so

$\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{2}^{*})\big{|}=\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{2}^{*})\big{|}$
(15)

where $\mathcal{P}_{\mathcal{M}}$ denote the projection of a vector onto
$\mathcal{M}$. Moreover,

$\displaystyle\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{2}^{*})\big{|}$
$\displaystyle\leq$
$\displaystyle\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}^{*}-x_{1})\big{|}+\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}-x_{2})\big{|}$
$\displaystyle+\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{2}-x_{2}^{*})\big{|}$
$\displaystyle\leq$ $\displaystyle
2\beta|x_{1}-x_{2}|_{2}\cdot\delta_{r}+\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot(x_{1}-x_{2})\big{|}\,.$
(16)

Let $k^{(1)},k^{(2)},\cdots,k^{(r)}\in\mathbb{Z}^{n}$ be the $K$-basis of
$\mathcal{M}$. Since $x_{1},x_{2}\in\mathcal{Z}_{\mathcal{M}}^{l}$, for each
$i\in\\{1,2,\ldots,r\\}$, there exists $l_{i}\in\mathbb{Z}$ such that
$\big{|}k^{(i)}\cdot\widetilde{\omega}(x_{1})+2\pi l_{i}\big{|}\leq\alpha_{r}$
and $\big{|}k^{(i)}\cdot\widetilde{\omega}(x_{2})+2\pi
l_{i}\big{|}\leq\alpha_{r}$ . Thus, we have

$\displaystyle\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot
k^{(i)}\big{|}=\big{|}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\cdot
k^{(i)}\big{|}$ $\displaystyle=$
$\displaystyle\big{|}k^{(i)}\cdot\widetilde{\omega}(x_{1})+2\pi
l_{i}-k^{(i)}\cdot\widetilde{\omega}(x_{2})-2\pi l_{i}\big{|}\leq
2\alpha_{r}\,.$

From Lemma 3.1 it follows:

$\big{|}\mathcal{P}_{\mathcal{M}}(\widetilde{\omega}(x_{1})-\widetilde{\omega}(x_{2}))\big{|}_{2}\leq
2rK^{r-1}\alpha_{r}\,.$ (17)

Combining (3) – (17), we get

$\beta\mu|x_{1}-x_{2}|_{2}^{2}\leq
4\beta|x_{1}-x_{2}|_{2}\cdot\delta_{r}+2rK^{r-1}\alpha_{r}\cdot|x_{1}-x_{2}|_{2}\,.$

That is

$|x_{1}-x_{2}|_{2}\leq\frac{4}{\mu}\delta_{r}+\frac{2rK^{r-1}}{\beta\mu}\alpha_{r}\,.$

∎

##  4 The Proof of Theorem 1

In this section, we will complete the proof of Theorem 1. Now, we have to make
a choice for free parameters $K,\alpha_{1},\cdots,\alpha_{n}$ and
$\delta_{1},\cdots,\delta_{n}$ in order to satisfy two important properties
which are crucial points in the following proof. First, there is no
intersection among the extended blocks with the same dimensional lattices.
More precisely,
$\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l})\bigcap\mathcal{Z}_{\mathcal{M}^{\prime}}^{l^{\prime}}=\emptyset$
for any $\mathcal{M}^{\prime}\neq\mathcal{M}$ with
$\text{dim}(\mathcal{M})=\text{dim}(\mathcal{M}^{\prime})=r$, and
$l^{\prime}\in\mathbb{Z}^{r}$, that is called the condition of nonoverlapping
of resonances [3]. Second, if action variables can leave the initial cylinder
in an exponentially long time, they must enter some resonant block associated
to a lower dimensional lattice.

Obviously, the condition of nonoverlapping of resonances is equivalent to the
following form:

$|k\cdot\widetilde{\omega}(x)+2\pi l_{0}|>\alpha_{r}\quad\text{for~{}
all}\,~{}x\in\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l}),~{}k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}~{}\text{and}~{}l_{0}\in\mathbb{Z}\,.$

In order to satisfy the above condition, we can make choice

$\displaystyle\alpha_{r}$
$\displaystyle=\Big{(}\frac{2}{\mu}\Big{)}^{r}r!K^{\frac{r(r-1)}{2}}\alpha_{1}\,,\quad
r\geq 2$ $\displaystyle\delta_{r}$
$\displaystyle=\frac{\alpha_{r}}{3K\beta}\,.$

Indeed, one can remark that for any
$x\in\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l})$,
there exists $x_{*}\in\mathcal{B}_{\mathcal{M}}^{l}$ such that
$|x-x_{*}|_{2}\leq\delta_{r}+\frac{4}{\mu}\delta_{r}+\frac{2rK^{r-1}}{\beta\mu}\alpha_{r}$
by the definition of the extended block and Proposition 3.1 (iv). Therefore,
for any
$x\in\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l}),~{}k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}$
, and $l_{0}\in\mathbb{Z}$, we have

$\displaystyle|k\cdot\widetilde{\omega}(x)+2\pi l_{0}|$
$\displaystyle\geq|k\cdot\widetilde{\omega}(x_{*})+2\pi
l_{0}|-|k\cdot(\widetilde{\omega}(x)-\widetilde{\omega}(x_{*}))|$
$\displaystyle>\alpha_{r+1}-K\beta|x-x_{*}|_{2}$
$\displaystyle\geq\alpha_{r+1}-K\beta\Big{(}\delta_{r}+\frac{4}{\mu}\delta_{r}+\frac{2rK^{r-1}}{\beta\mu}\alpha_{r}\Big{)}\,.$

Thus the nonoverlapping condition of resonances is satisfied only if
$\alpha_{r+1}-K\beta\big{(}\delta_{r}+\frac{4}{\mu}\delta_{r}+\frac{2rK^{r-1}}{\beta\mu}\alpha_{r}\big{)}\geq\alpha_{r}$
through the above choices. By means of the nonoverlapping condition of
resonances, we can prove that

$\Big{|}1-e^{ik\cdot\widetilde{\omega}(x)}\Big{|}\geq\frac{2\alpha_{r}}{\pi}\quad\text{for~{}all}\,~{}x\in\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l}),~{}k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}\,.$
(18)

For this purpose, we can choose $l=l(\widetilde{\omega},k)\in\mathbb{Z}$ such
that $\Big{|}\frac{k\cdot\widetilde{\omega}(x)+2\pi
l}{2}\Big{|}\in\big{[}0,\frac{\pi}{2}\big{]}$. Thus,

$\Big{|}1-e^{ik\cdot\widetilde{\omega}(x)}\Big{|}=2\sin\bigg{|}\frac{k\cdot\widetilde{\omega}(x)+2\pi
l}{2}\bigg{|}\geq\frac{4}{\pi}\bigg{|}\frac{k\cdot\widetilde{\omega}(x)+2\pi
l}{2}\bigg{|}\geq\frac{2}{\pi}\alpha_{r}\,.$

When choosing
$\rho_{1}^{(r)}=\min\big{(}\frac{\alpha_{r}}{4K\beta},1\big{)}<\delta_{r}$ and
$\rho_{2}^{(r)}=\widetilde{\sigma}_{2}$, Normal Form Lemma can be applied in
the domain
$\mathcal{V}_{\rho_{1}^{(r)}/2}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l})\times\mathcal{V}_{\rho_{2}^{(r)}/2}(T^{n})$
with $\alpha=\frac{2}{\pi}\alpha_{r}$, if the following conditions are
satisfied:

$\displaystyle||Df||_{G^{(r)},\,\frac{\rho^{(r)}}{2},\,c^{(r)}}$
$\displaystyle\leq\frac{2}{\pi}\cdot\frac{\alpha_{r}\rho_{1}^{(r)}}{\widetilde{C}AK\rho_{2}^{(r)}}\,,$
(19) $\displaystyle\rho_{1}^{(r)}$
$\displaystyle\leq\widetilde{\sigma}_{1}\,,$ (20)

where
$G^{(r)}=\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l},~{}c^{(r)}=\frac{\rho_{1}^{(r)}}{\rho_{2}^{(r)}}$
.

On the other hand, by Lemma 2.3 we have
$||Df||_{G^{(r)},\,\frac{\rho^{(r)}}{2},\,c^{(r)}}\leq\frac{2}{\rho_{2}^{(r)}}||f||_{\mathcal{G}_{\gamma},\,\widetilde{\sigma}}=\frac{2}{\rho_{2}^{(r)}}\tilde{\epsilon}$.
Therefore, in order to satisfy (19), we require

$\tilde{\epsilon}\leq\frac{\alpha_{r}\rho_{1}^{(r)}}{\widetilde{C}AK\pi}\,.$
(21)

According to Normal Form Lemma, there exists a symplectic transformation
$\Phi:\mathcal{D}_{\rho^{(r)}/4}(G^{(r)})\rightarrow\mathcal{D}_{\rho^{(r)}/2}(G^{(r)})$
such that the conjugate symplectic map $T_{\gamma}^{\prime}=\Phi^{-1}\circ
T_{\gamma}\circ\Phi$ is generated by the analytic function
$F^{\prime}=F_{0}+Z+R$ with $Z\in\mathcal{R}(\mathcal{M},K)$ and

$||DR||_{G^{(r)},\,\frac{\rho^{(r)}}{4},\,c^{(r)}}\leq
3e^{-\frac{K\rho_{2}^{(r)}}{24}}\frac{2}{\rho_{2}^{(r)}}\cdot\tilde{\epsilon}\,.$

Claim 1. Under the above conditions, denote by $\tau_{+}$ and $\tau_{-}$ the
possibly times of escape of $x(t)$ from $\mathcal{Z}_{\mathcal{M}}^{l}$ at
positive and negative times respectively. Then for any
$x_{0}\in\mathcal{B}_{\mathcal{M}}^{l}$ , one has
$x(t)\in\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x_{0})$ if
$|t|\leq\min(T^{(r)},\,\tau_{+},\,-\tau_{-})$ with
$T^{(r)}=\frac{\rho_{1}^{(r)}\rho_{2}^{(r)}e^{\frac{K\rho_{2}^{(r)}}{24}}}{2^{7}\tilde{\epsilon}}$
.

Consider the action variables of $T_{\gamma}^{\prime}$, it is given by

$\hat{a}=a-\Big{(}\frac{\partial
Z(\hat{a},\varphi)}{\partial\varphi}+\frac{\partial
R(\hat{a},\varphi)}{\partial\varphi}\Big{)}\,,$

and consider a auxiliary system

$\tilde{a}=a-\frac{\partial Z(\hat{a},\varphi)}{\partial\varphi}\,.$

After iterating $t$ times, we have $\tilde{a}(t)\in\Pi_{\mathcal{M}}(a(0))$,
and

$|a(t)-\tilde{a}(t)|_{2}\leq|t|\cdot\Big{|}\Big{|}\frac{\partial
R(\hat{a},\varphi)}{\partial\varphi}\Big{|}\Big{|}_{G^{(r)},\,\frac{\rho^{(r)}}{4},\,1}\leq
T^{(r)}||DR||_{G^{(r)},\,\frac{\rho^{(r)}}{4},\,c^{(r)}}\leq\frac{\rho_{1}^{(r)}}{2^{4}}\,.$

Thus,

$\displaystyle\text{dist}(x(t),\Pi_{\mathcal{M}}(x(0)))$
$\displaystyle\leq|x(t)-a(t)|_{2}+|a(t)-\tilde{a}(t)|_{2}+\text{dist}(\tilde{a}(t),\Pi_{\mathcal{M}}(x(0)))$
$\displaystyle\leq\frac{\rho_{1}^{(r)}}{2^{8}}+\frac{\rho_{1}^{(r)}}{2^{4}}+\frac{\rho_{1}^{(r)}}{2^{8}}$
$\displaystyle\leq\frac{\rho_{1}^{(r)}}{4}\,.$

That is $x(t)\in\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x_{0})$ if
$|t|\leq\min(T^{(r)},\,\tau_{+},\,-\tau_{-})$.

Claim 2. Assume $\tau_{+},-\tau_{-}<T^{(r)}$ and there exists $t_{*}$
satisfying $x(t_{*})\in\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x_{0})$, but
$x(t_{*}+1)\notin\mathcal{C}_{\mathcal{M},\,\delta_{r}}^{l}(x_{0})$, then
$x(t_{*}+1)\in\mathcal{B}_{\mathcal{M}^{\prime}}^{l^{\prime}}$ with
dim$(\mathcal{M}^{\prime})<r$ and some $l^{\prime}$.

Note that

$\displaystyle|x(t_{*}+1)-x(t_{*})|_{2}$
$\displaystyle\leq|x(t_{*}+1)-a(t_{*}+1)|_{2}+\Big{|}\Big{|}\frac{\partial
Z(\hat{a},\varphi)}{\partial\varphi}\Big{|}\Big{|}_{G^{(r)},\,\frac{\rho^{(r)}}{4},\,1}$
$\displaystyle\quad+\Big{|}\Big{|}\frac{\partial
R(\hat{a},\varphi)}{\partial\varphi}\Big{|}\Big{|}_{G^{(r)},\,\frac{\rho^{(r)}}{4},\,1}+|a(t_{*})-x(t_{*})|_{2}$
$\displaystyle\leq\frac{\rho_{1}^{(r)}}{2^{8}}+2||Df||_{G^{(r)},\,\frac{\rho^{(r)}}{2},\,c^{(r)}}+3||Df||_{G^{(r)},\,\frac{\rho^{(r)}}{2},\,c^{(r)}}+\frac{\rho_{1}^{(r)}}{2^{8}}$
$\displaystyle\leq\frac{\rho_{1}^{(r)}}{2^{7}}+5\cdot\frac{2}{\rho_{2}^{(r)}}\tilde{\epsilon}$

where we have used Normal Form Lemma. Because of $\widetilde{C}\geq
51,~{}A\geq 1,~{}\alpha_{r}<1$ and (21), we have

$\frac{2}{\rho_{2}^{(r)}}\tilde{\epsilon}\leq\frac{\rho_{1}^{(r)}}{6*51}\cdot\frac{1}{K\rho_{2}^{(r)}}\leq\frac{\rho_{1}^{(r)}}{6*51}\,,$

when $K\geq\frac{1}{\rho_{2}^{(r)}}=\frac{1}{\widetilde{\sigma_{2}}}$ . Thus,
$|x(t_{*}+1)-x(t_{*})|_{2}\leq\frac{\rho_{1}^{(r)}}{4}<\frac{\delta_{r}}{4}$ .
That means
$x(t_{*}+1)\in\mathcal{V}_{\delta_{r}}(\mathcal{B}_{\mathcal{M},\,\delta_{r}}^{l})$.
At this time there are only two possible cases:
$x(t_{*}+1)\in\mathcal{Z}_{\mathcal{M}}^{l^{\prime}}~{}\text{for
some}~{}l^{\prime}\in\mathbb{Z}^{r}$ or $x(t_{*}+1)\notin\mathcal{Z}_{r}^{*}$
by the nonoverlapping condition of resonances. However, for any
$l^{\prime}\neq l$, $x(t_{*}+1)\notin\mathcal{Z}_{\mathcal{M}}^{l^{\prime}}$.
In fact, because $x(t_{*})\in\mathcal{Z}_{\mathcal{M}}^{l}$, we have
$\big{|}k^{(j)}\cdot\widetilde{\omega}(x(t_{*}))+2\pi
l_{j}\big{|}\leq\alpha_{r},~{}j=1,\ldots,r$. Thus, for any
$l_{j}^{\prime}\in\mathbb{Z}$ and $l_{j}^{\prime}\neq l_{j}$,

$\displaystyle\big{|}k^{(j)}\cdot\widetilde{\omega}(x(t_{*}+1))+2\pi
l_{j}^{\prime}\big{|}$
$\displaystyle=\big{|}k^{(j)}\cdot\big{(}\widetilde{\omega}(x(t_{*}+1))-\widetilde{\omega}(x(t_{*}))\big{)}+k^{(j)}\cdot\widetilde{\omega}(x(t_{*}))+2\pi
l_{j}$ $\displaystyle\quad+2\pi l_{j}^{\prime}-2\pi l_{j}\big{|}$
$\displaystyle\geq 2\pi-\alpha_{r}-K\beta\cdot\frac{\delta_{r}}{4}$
$\displaystyle>1.$

Therefore $x(t_{*}+1)\notin\mathcal{Z}_{r}^{*}$. Because of the property (iii)
in Proposition 3.1, one has $x(t_{*}+1)\in\mathcal{B}_{\mathcal{M}^{\prime}}$
with dim$(\mathcal{M}^{\prime})<r$. It is shown that $x(t)$ enter some
resonant block associated to lower dimensional lattice, after going out of the
original cylinder only through its base during $|t|<T$.

The stability estimates now apply to all blocks simultaneously, if

$\displaystyle\tilde{\epsilon}=\frac{M}{\beta}\cdot\epsilon\leq\frac{\alpha_{1}\rho_{1}^{(1)}}{\widetilde{C}AK\pi}\,,$
(22) $\displaystyle\rho_{1}^{(n)}\leq\widetilde{\sigma}_{1}\,,$ (23)
$\displaystyle\beta\leq\min_{1\leq i\leq n}\\{\alpha_{i}\\}\,,$ (24)
$\displaystyle K\geq\frac{1}{\widetilde{\sigma}_{2}}=\frac{1}{\sigma_{2}}\,.$
(25)

The (22) – (23) are satisfied if we require

$\displaystyle\epsilon\leq\frac{\alpha_{n}^{2}}{4\pi(21n+30)\big{[}(\frac{2M}{m})^{n}\cdot
n!\big{]}^{2}K^{n^{2}-n+2}\cdot M}\,,$ (26) $\displaystyle\alpha_{n}\leq
4KM\sigma_{1}\,.$ (27)

On the other hand, in order to keep as small as possible for the diameter of
the cylinders, at least the order of $K^{-1}$, it is convenient to put
$\alpha_{n}=4MK^{-n}\sigma_{1}$. Due to $\sigma_{1}<\frac{1}{4M}$, so
$\alpha_{n}<1$. Finally, let $\beta=\min\limits_{1\leq i\leq
n}\\{\alpha_{i}\\}$ and $K=\epsilon^{-\frac{1}{b}}$ with $b=2(n^{2}+n+2)$.
Then, $K\geq\frac{1}{\sigma_{2}}$ when $\epsilon\leq\sigma_{2}^{b}$.

From the above discussions, it is clear that for any initial value $x_{0}$,
the adapted normal form can be constructed in the corresponding extended
resonant block. The normal form provides the confinement of the action $x(t)$
in $\Pi_{\mathcal{M}}(x_{0})$. Moreover, $x(t)$ must enter one of the other
resonant block with lower dimensional multiplicity after out of the previous
one, arrives in the nonresonant block at worst, where it stops. Thus, the
stability radius $\widetilde{\triangle}$ satisfies

$\widetilde{\triangle}\leq(n-1)(\frac{4M}{m}\delta_{n}+\frac{2nK^{n-1}M}{\beta
m}\alpha_{n})+\widetilde{\triangle}_{0}\,,$

where $\widetilde{\triangle}_{0}$ denotes the deviation of the action
variables when them get into the nonresonant block and $|t|\leq T^{(1)}$. This
number can be estimated directly as follows.

According to Proposition 3.1 (i), for any $x\in\mathcal{B}_{0}$,
$\big{|}k\cdot\widetilde{\omega}(x)+2\pi l\big{|}>\alpha_{1}$ for all
$k\in\mathbb{Z}_{K}^{n}\backslash\\{0\\}$ and $l_{0}\in\mathbb{Z}$. Then we
can prove

$\Big{|}1-e^{ik\cdot\widetilde{\omega}(x)}\Big{|}\geq\frac{2}{\pi}\alpha_{1}\quad\text{for~{}all}\,~{}x\in\mathcal{B}_{0},~{}k\in\mathbb{Z}_{K}^{n}\backslash\\{0\\}\,.$

by the same technique that be used in the proof of (18). Furthermore, applying
Lemma 2.1 with
$G=\mathcal{B}_{0},~{}\rho_{1}=\rho_{1}^{(1)},~{}\alpha=\frac{2}{\pi}\alpha_{1}$,
and $\mathcal{M}=\\{0\\}$, we have

$\Big{|}1-e^{ik\cdot\widetilde{\omega}(x)}\Big{|}\geq\frac{\alpha_{1}}{\pi}\quad\text{for~{}all}\,~{}x\in\mathcal{V}_{\rho_{1}^{(1)}}(\mathcal{B}_{0}),~{}k\in\mathbb{Z}_{K}^{n}\backslash\\{0\\}\,.$

Now, we can apply Normal Form Lemma with
$G=\mathcal{B}_{0},~{}\rho=\frac{\rho^{(1)}}{2},~{}\alpha=\frac{\alpha_{1}}{\pi}$,
and $\mathcal{M}=\\{0\\}$ under the condition (26). Then there exists a real
analytic canonical transformation
$\Phi:\mathcal{D}_{\rho^{(1)}/4}(\mathcal{B}_{0})\rightarrow\mathcal{D}_{\rho^{(1)}/2}(\mathcal{B}_{0})$
such that $T_{\gamma}^{\prime}=\Phi^{-1}\circ T_{\gamma}\circ\Phi$ is
generated by the analytic function
$F^{\prime}(\hat{a},\varphi)=F_{0}(\hat{a})+Z(\hat{a},\varphi)+R(\hat{a},\varphi)$
with $Z\in\mathcal{R}(\mathcal{M},K)$. Due to $\mathcal{M}=\\{0\\}$, $Z$ only
depends on the action variables, and the system becames

$\left\\{\begin{aligned} \hat{a}&=a-\frac{\partial
R(\hat{a},\varphi)}{\partial\varphi}\\\ \hat{\varphi}&=\varphi+\frac{\partial
F_{0}(\hat{a})}{\partial\hat{a}}+\frac{\partial
Z(\hat{a})}{\partial\hat{a}}+\frac{\partial
R(\hat{a},\varphi)}{\partial\hat{a}}\,.\end{aligned}\right.$

After iterating $t$ times with $|t|\leq T^{(1)}$,

$\displaystyle|a(t)-a(0)|_{2}$
$\displaystyle\leq|t|\cdot\Big{|}\Big{|}\frac{\partial
R}{\partial\varphi}\Big{|}\Big{|}_{\mathcal{B}_{0},\,\frac{\rho^{(1)}}{4},\,1}$
$\displaystyle\leq T^{(1)}\cdot
3e^{-\frac{K\rho_{2}^{(1)}}{24}}||Df||_{\mathcal{B}_{0},\,\frac{\rho^{(1)}}{2},\,c^{(1)}}$
$\displaystyle\leq\frac{\rho_{1}^{(1)}}{2^{4}}\,.$

Thus,
$|x(t)-x(0)|_{2}\leq|x(t)-a(t)|_{2}+|a(t)-a(0)|_{2}+|a(0)-x(0)|_{2}\leq\frac{\rho_{1}^{(1)}}{4}\leq\delta_{n}$
. That means $\widetilde{\triangle}_{0}\leq\delta_{n}$ and

$\widetilde{\triangle}\leq n(\frac{4M}{m}\delta_{n}+\frac{2nK^{n-1}M}{\beta
m}\alpha_{n})\,.$

Therefore, we have

$|x(t)-x(0)|_{2}\leq n(\frac{4M}{m}\delta_{n}+\frac{2nK^{n-1}M}{\beta
m}\alpha_{n})\quad\text{for~{}all}\,~{}x(0)\in\mathcal{G}_{\gamma}~{}\text{and}\,~{}|t|\leq
T^{(1)}\,.$

Note that $I=\gamma x$, so

$|I(t)-I(0)|_{2}\leq\gamma\widetilde{\triangle}\leq
n\Big{(}\frac{16M}{3m}+\frac{8nM}{m}\Big{)}\sigma_{1}\cdot\epsilon^{\frac{1}{b}}:=c_{0}\epsilon^{\frac{1}{b}}:=\triangle\,.$

where $c_{0}=\frac{8nM}{3m}(3n+2)\sigma_{1}$.

Combining the above discussion, we can choose the stability time
$T=\min\limits_{1\leq i\leq n}\\{T^{(i)}\\}=T^{(1)}$. Precisely,

$T=\frac{\rho_{1}^{(1)}\rho_{2}^{(1)}e^{\frac{K\rho_{2}^{(1)}}{24}}}{2^{7}\tilde{\epsilon}}=\frac{\alpha_{1}\sigma_{2}}{2^{9}\epsilon
KM}\cdot
e^{\frac{\sigma_{2}}{24}\epsilon^{-\frac{1}{b}}}=T_{0}\epsilon^{-\frac{3}{4}}e^{\frac{\sigma_{2}}{24}\epsilon^{-\frac{1}{b}}}\,,$

with $T_{0}=\frac{(\frac{m}{2M})^{n}\sigma_{1}\sigma_{2}}{2^{7}n!}$ . And the
perturbation $\epsilon\leq\min(\epsilon_{0},\,\sigma_{2}^{b})$, where

$\epsilon_{0}=\frac{M^{2}\cdot\sigma_{1}^{4}}{\pi^{2}(21n+30)^{2}\big{[}(\frac{2M}{m})^{n}\cdot
n!\big{]}^{4}}\,.$

Finally, in order to prevent action variables from going out of $\mathcal{G}$,
we restrict $I_{0}\in\mathcal{G}-\triangle$.

##  5 Application

An application of the above theorem gives the exponential stability of a
nearly integrable symplectic map with a small twist, which often comes from
numerical discretization of Hamiltonian systems. Consider a one-parameter
family of symplectic map $\mathcal{C}_{s}$ with the parameter $s$ satisfying
$0<s\leq 1$ and the analytic generating function
$H_{s}=sH_{0}(\hat{I})+sh(\hat{I},\theta)$. The small twist map is given by

$\displaystyle\hat{I}$ $\displaystyle=I-s\frac{\partial
h(\hat{I},\theta)}{\partial\theta}~{},$ $\displaystyle\hat{\theta}$
$\displaystyle=\theta+s\frac{\partial
H_{0}(\hat{I})}{\partial\hat{I}}+s\frac{\partial
h(\hat{I},\theta)}{\partial\hat{I}}~{}.$

The result can ba stated as follows.

Theorem 5.1 Let $H_{s}$ be analytic in $\mathscr{D}_{\sigma}(\mathcal{G})$,
where $\mathcal{G}$ is an open bounded domain of $\mathbb{R}^{n}$,
$\sigma=(\sigma_{1},\sigma_{2})$ is positive, and $\omega(\hat{I})=\partial
H_{0}(\hat{I})$ satisfying (1). Consider the above small twist symplectic map
$\mathcal{C}_{s}$ defined on $\mathcal{G}\times T^{n}$. If

$\epsilon=||h||_{\mathcal{G},\,\sigma}\leq\min(\epsilon_{0},\,\sigma_{2}^{b})\,,$
(28)

where

$\epsilon_{0}=\frac{M^{2}\cdot\sigma_{1}^{4}}{\pi^{2}(21n+30)^{2}\big{[}(\frac{2M}{m})^{n}\cdot
n!\big{]}^{4}}\,,$

$b=2(n^{2}+n+2)$ and $\sigma_{1}<\frac{1}{4M}$ . Then for any $0<s\leq 1$, the
symplectic map $\mathcal{C}_{s}$ satisfies

$|I(t)-I_{0}|_{2}\leq\triangle\quad\mbox{for}\quad|t|\leq
T\quad\mbox{and}\quad I_{0}\in\mathcal{G}-\triangle\,,$

where $(I(t),\theta(t))=\mathcal{C}^{t}(I_{0},\theta_{0})$ ($t$ is viewed as
iterative times),

$\displaystyle\triangle$ $\displaystyle=c_{0}\epsilon^{\frac{1}{b}}\quad$ with
$\displaystyle c_{0}=\frac{8nM}{3m}(3n+2)\sigma_{1}\,,$ $\displaystyle T$
$\displaystyle=T_{0}\epsilon^{-\frac{3}{4}}e^{c_{1}\epsilon^{-\frac{1}{b}}}\quad$
with $\displaystyle
T_{0}=\frac{(\frac{m}{2M})^{n}\sigma_{1}\sigma_{2}}{2^{7}n!}\quad\mbox{and}\quad
c_{1}=\frac{\sigma_{2}}{24}\,.$

Here we only outline the proof of the above theorem since it is almost the
same as the proof of Theorem 1. First, in this case, the small denominators
became $1-e^{ik\cdot s\widetilde{\omega}(x)}$. Therefore, in order to
construct the resonant normal form with respect to the $K$-lattice
$\mathcal{M}$, we restrict to the subset

$G_{s}=\big{\\{}x\in\mathcal{G}_{\gamma}:~{}|1-e^{ik\cdot
s\widetilde{\omega}(x)}|\geq
s\alpha\,,\quad\forall~{}\,k\in\mathbb{Z}_{K}^{n}\backslash\mathcal{M}\big{\\}}\,.$

The corresponding Normal Form Lemma holds if the quantity are
$sf,\,sM,\,sm,\,s\alpha$ and $s\beta$ instead of $f,\,M,\,m,\,\alpha$ and
$\beta$ respectively. That means if
$s||Df||_{G,\,\rho,\,c}\leq\frac{s\alpha\,\rho_{1}}{\widetilde{C}AK\rho_{2}}\,,$
then there exists a real analytic canonical transformation
$\Phi:\mathcal{D}_{\frac{\rho}{2}}(G)\rightarrow\mathcal{D}_{\rho}(G)$ such
that the conjugate symplectic map $T_{\gamma}^{\prime}=\Phi^{-1}\circ
T_{\gamma}\circ\Phi:\mathcal{D}_{\frac{\rho}{2}}(G)\rightarrow\mathcal{D}_{\rho}(G)$
is generated by the analytic function $F^{\prime}=sF_{0}+sZ+sR$ with
$Z\in\mathcal{R}(\mathcal{M},K)$ and $R$ exponential small. Moreover, the same
estimates about $Z$ and $R$ hold like in Lemma 2.2.

On the other hand, the geometric construction also needs some modifications.
Precisely, the parameter $s$ enters the definitions of resonant manifold,
zone, and block. $\widetilde{\omega}(x)$ and $\alpha_{r}$ should be replaced
by $s\widetilde{\omega}(x)$ and $s\alpha_{r}$ respectively. Thus we obtain the
corresponding resonant manifold, zone, and block. We remark that the arguments
of the proposition 3.1 still hold in this case.

Finally, we make the same choices about the parameters
$\alpha_{r},\delta_{r}~{}(r=1,\ldots,n)$ and $K$ as before, and the desired
estimates can be derived. The details are omitted here.

In the following contents, we will discuss the exponential stability of
symplectic integrator which applied to the integrable Hamiltonian system by
the Theorem 5.1. Integrable Hamiltonian system is a very important class of
dynamic system. In general it possesses many enough first integrals.
Therefore, it exhibits regular dynamic behavior which corresponds to periodic
and quasi-periodic motions in phase space through action-angle variables.
However, in many cases the action-angle variables may not be known explicitly.
Then it is difficult to compute solutions of the given integrable Hamiltonian
system, and numerical integration is necessary.

After the pioneering work of Channel (1983), Feng Kang (1985, 1986) and Ruth
(1983), the symplectic integrator has become a widely interested subject on
the problem of numerically solving Hamiltonian systems. Extensive computer
experimentation, by some typical models of Hamiltonian systems, has shown the
overwhelming superiority of symplectic algorithms over the conventional non-
symplectic ones, especially in simulating the global and structural dynamic
behavior of the systems (e.g. see [14] and [11]). The symplectic algorithm,
which applied to integrable Hamiltonian system, may be characterized as a
perturbation of the phase flow of the integrable system. Here the smallness of
the perturbation is described by the time-step size of the algorithm which
also enters into the frequency map of the integrable system. Therefore
numerical stability problem arises.

There has been recently some nice work about the numerical analysis of
symplectic algorithms for Hamiltonian systems, for example, by Benettin &
Giorgilli [4], Hairer & Lubich [13], Shang [20], and Stoffer [22]. Stoffer
proved the numerical solutions is integrable up to a remainder which is
exponential small with respect to the step-size when a symplectic integrator
is applied to a integrable system. However, the result requires that the
initial frequency satisfying the strong non-resonance condition. In addition,
for non-resonance time step-size, Shang obtained the existence result of
numerical invariant tori of symplectic algorithms.

We consider a integrable Hamiltonian system (usually not given in action-angle
variables)

$\dot{p}=-\frac{\partial H(p,q)}{\partial q},\quad\dot{q}=\frac{\partial
H(p,q)}{\partial p}$ (29)

and apply to it a symplectic algorithm $G_{H}^{h}$ of order $r$ with step size
$h$. For an overview on the symplectic integrators see the book of Haire,
Lubich and Wanner [14]. Because of Arnold-Liouville theorem, there exists a
symplectic transformation $\Psi:(I,\theta)\rightarrow(p,q)$ such that the new
Hamiltonian $\mathcal{H}(I)=H\circ\Psi(I,\theta)$, that only depends on the
action variables, $(I,\theta)\in\mathcal{G}\times T^{n}$. Here we assume $H$
and $\mathcal{H}$ is analytic in $\mathcal{D}_{\rho}(\mathcal{G})$ and
$\mathcal{V}_{\rho_{1}}(\mathcal{G})$ respectively. And
$\omega(I)=\frac{\partial\mathcal{H}(I)}{\partial I}$ satisfies the condition
(1). In the action-angle variables, the equation (29) takes the simple form

$\dot{I}=0,\quad\dot{\theta}=\omega(I)=\frac{\partial\mathcal{H}(I)}{\partial
I}.$ (30)

The symplectic integrator $G_{H}^{h}$ becames
$\widetilde{G}_{H}^{h}=\Psi^{-1}\circ G_{H}^{h}\circ\Psi$.

Lemma 5.1 [20] There exists a function $f^{h}$ which depends on the time step
$h$ such that it is well-defined and real analytic in the domain
$\mathcal{D}_{\frac{\rho}{4}}(\mathcal{G})$ for $h\in[0,\delta]$ with $\delta$
being a sufficiently small positive number so that
$\widetilde{G}_{H}^{h}:(I,\theta)\rightarrow(I^{\prime},\theta^{\prime})$ can
be expressed by $f^{h}$ as follows:

$I^{\prime}=I-h^{r+1}\frac{\partial
f^{h}(I^{\prime},\theta)}{\partial\theta},\quad\theta^{\prime}=\theta+h\omega(I^{\prime})+h^{r+1}\frac{\partial
f^{h}(I^{\prime},\theta)}{\partial I^{\prime}}.$ (31)

Moreover, there exists $L$ independent on $h$ such that $||f^{h}||\leq L$.

Now the Theorem 5.1 can be applied to $\mathcal{C}_{h}=\widetilde{G}_{H}^{h}$
if $h^{r}f^{h}$ satisfies the estimate (28) with $\sigma=\frac{\rho}{4}$.
Therefore we have the following result.

Theorem 5.2 Under the above assumption on $\mathcal{H}$, apply a symplectic
method $G_{H}^{h}$ of order $r$ to the equation (29). Let
$\widetilde{G}_{H}^{h}$ generate an orbit
$(I_{1},\theta_{1}),(I_{2},\theta_{2}),\ldots$ with any initial value
$(I_{0},\theta_{0})$ in action-angle variables. Then there are positive
constants $h_{0},c_{0},c_{1},T_{0}$ such that for all $h\leq h_{0}$, the
following estimates hold

$|I_{m}-I_{0}|_{2}\leq c_{0}h^{\frac{r}{b}}\,,$

for all $m$ with $mh\leq T_{0}h^{-\frac{3}{4}r}e^{c_{1}h^{-\frac{r}{b}}}$ and
$b=2(n^{2}+n+2)$.

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