# A KAM theorem of symplectic algorithms for nearly integrabel Hamiltonian
systems

Zaijiu Shang , Yang Xu

###### Abstract

In this paper we prove a KAM-like theorem of symplectic algorithms for nearly
integrable Hamiltonian systems which generalises the result of [1] and [5] for
the case of integrable systems.

††Zaijiu Shang: Institute of Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100080, China; e-mail<EMAIL_ADDRESS>
Yang Xu: School of Mathematical Sciences, Fudan University, Shanghai 200433,
China; e-mail<EMAIL_ADDRESS>
Keywords. nearly integrable, Symplectic algorithm, KAM theorem, Rüssmann’s
non-degeneracy

###### Contents

  1. 1 Introduction
  2. 2 Main theorem
  3. 3 Proof of Theorem 2.1
    1. 3.1 One-step Iterative Analysis
    2. 3.2 Proof of the Iteration Lemma
    3. 3.3 Proof of convergence
    4. 3.4 Measure Estimation
  4. 4 Kolmogorov’s non-degeneracy condition

##  1 Introduction

In [1] the author established a KAM theorem of symplectic algorithms for
integrable Hamiltonian systems and left the question of generalization to
nearly integrable systems open. In this paper we will solve it. The notations
of [1] will be still used.

Consider the nearly integrable Hamiltonian system :

$\dot{p}=-\frac{\partial H^{\epsilon}}{\partial q}(p,q),\
\dot{q}=\frac{\partial H^{\epsilon}}{\partial p}(p,q),\ (p,q)\in D,$ (1.1)

where $H^{\epsilon}(p,q)=H^{0}(p)+\epsilon H(p,q,\epsilon)$ is the Hamiltonian
function, $D\subseteq{\mathbb{R}}^{n}\times\mathbb{T}^{n}$ is a bounded
connected open domain, and the dot is the derivative with respect to time $t$.
For such nearly integrable Hamiltonian system, if the perturbation is small
enough and the Hamiltonian function is sufficiently smooth, then there exist
invariant tori, which is what we know. In this paper, we will use the method
of discretization of symplectic algorithm to show that the system after the
discretization of symplectic algorithm also has invariant tori. Compared with
the invariant tori of the nearly integrable Hamiltonian system, there is only
slight deformation between them. Without loss of generality, we can assume
$\lvert H\rvert_{D}\leq M_{1}$, where $\lvert\cdot\lvert_{D}$ denotes the
supremum norm on $D$, $M_{1}$ is a constant, and we can ignore the parameter
expression of $\epsilon$.

Denote $\omega$ as the derivative of $H^{0}$, $\omega(\xi)=\frac{\partial
H^{0}}{\partial\xi}(\xi)$, which is defined on $V_{\kappa}$, where
$V\subset\mathbb{R}^{n}$ is a bounded connected domain,
$V_{\kappa}=V+\kappa:=\bigcup\limits_{b\in
V}\\{\xi\in\mathbb{C}^{n}:\lvert\xi-b\rvert_{2}<\kappa\\}\subseteq\mathbb{C}^{n}$,
and $\kappa$ is a constant.

Use symplectic algorithm to discretize the Hamiltonian system (1.1). By Lemma
3.1 and Lemma 3.3 in [1], there is a function $P$, which depends on time step
$t$, and it is well defined and real analytic on
$D_{\frac{r}{4},\frac{s}{4}}$, such that if time step $t$ is small enough,
$0<t<\delta$, where $\delta$ is a small enough constant, then the symplectic
differential form of the system (1.1) after discretization by symplectic
algorithm can be expressed as $G_{H^{\epsilon}}^{t}:(p,q)\to(\hat{p},\hat{q})$

$G_{H^{\epsilon}}^{t}:\begin{cases}\hat{p}=p-t\epsilon\frac{\partial
S}{\partial q}(\hat{p},q)-t^{\alpha+1}\frac{\partial{P}}{\partial
q}(\hat{p},q),\\\ \hat{q}=q+t\omega+t\epsilon\frac{\partial
S}{\partial\hat{p}}(\hat{p},q)+t^{\alpha+1}\frac{\partial{P}}{\partial\hat{p}}(\hat{p},q),\end{cases}$
(1.2)

where $(\hat{p},q)\in D_{\frac{r}{4},\frac{s}{4}}$, and $\alpha$ is a positive
constant. Here $S$ is a generating function,
$S(p,q,t)=\sum\limits_{k=0}^{\infty}(S)_{k}(p,q)t^{k}=H^{1}(p,q)+(S)_{1}(p,q)t+(S)_{2}(p,q)t^{2}+\dots$,
and $D_{r,s}$ refers to

$D_{r,s}=\\{p:\inf\limits_{p^{*}\in V}\lvert p-p^{*}\rvert<r\\}\times\\{q:\Re
q\in\mathbb{T}^{n},\lvert\Im
q\rvert<s\\}\subset\mathbb{C}^{n}\times\mathbb{C}^{n}.$

By Lemma 3.1 and Lemma 3.3 in [1], $\big{\lvert}\frac{\partial
P}{\partial{\hat{p}}}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq M_{2}$,
$\big{\lvert}\frac{\partial P}{\partial
q}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq M_{2}$, where
$\lvert\cdot\lvert_{\frac{r}{4},\frac{s}{4}}$ denotes the supremum norm on
$D_{\frac{r}{4},\frac{s}{4}}$, and $M_{2}$ is a constant. Similar to [1],
fixed $(\hat{p_{0}},q_{0})\in D$, let $P(\hat{p_{0}},q_{0})=0$, for
$(\hat{p},q)\in D_{\frac{r}{4},\frac{s}{4}}$, then we have $\lvert
P\rvert_{\frac{r}{4},\frac{s}{4}}\leq 2nM_{2}l_{*}$, where $l_{*}$ is an upper
bound of the length of the shortest curves from $(\hat{p_{0}},q_{0})$ to
$(\hat{p},q)$ in $D_{\frac{r}{4},\frac{s}{4}}$. And, $\lvert
G_{H^{\epsilon}}^{t}(p,q)-g_{H^{\epsilon}}^{t}(p,q)\rvert\leq
M_{3}t^{\alpha+1}$, where $g_{H^{\epsilon}}^{t}$ is the phase flow of the
nearly integrable Hamiltonian system (1.1), and $M_{3}$ is a constant that
does not depend on $t$.

For the nearly integrable Hamiltonian system (1.1), there exists a generating
function $\widetilde{S}$ such that its invariant tori can be found by the
following system:

$\hat{p}=p-t\epsilon\frac{\partial\widetilde{S}}{\partial q}(\hat{p},q),\
\hat{q}=q+t\omega+t\epsilon\frac{\partial\widetilde{S}}{\partial\hat{p}}(\hat{p},q),$
(1.3)

and for the time step $t$ in the common area, we have
$S(p,q,t)-\widetilde{S}(p,q,t)\sim o(t^{\alpha})$, let us denote it as $\lvert
S-\widetilde{S}\rvert\leq M_{4}t^{\alpha}$, where $M_{4}$ is a constant that
does not depend on $t$.

In this paper, we assume that $\omega$ satisfies the Rüssmann’s non-degeneracy
condition.

######  Remark 1.1.

$\omega$ satisfies the Rüssmann’s non-degeneracy condition means that
$\omega\in\\{\omega(\xi):\xi\in V\\}$ does not lie in any hyperplane that
passes through the origin, i.e. $\langle\omega,y\rangle\not\equiv 0$ for all
$y\in\mathbb{R}^{n}\setminus\\{0\\}$. Xu, You and Qiu gave an expression of
the Rüssmann’s non-degeneracy condition for analytic cases(see Remark 3.1 in
[2]), that is, if $\omega$ satisfies the Rüssmann’s non-degeneracy condition,
then there exists an integer $\bar{n}>0$, such that

$rank\big{\\{}\partial^{i}_{\xi}\omega(\xi):|i|\leq\bar{n}\big{\\}}=n,\
\forall\xi\in V.$ (1.4)

And, by Rüssmann (Lemma 18.2 in [4]), there are numbers
$\bar{n}(V)\in\mathbb{N}$ and $\beta_{0}(V)>0$ such that

$\min_{\xi\in V}\max_{0\leq v\leq\bar{n}}|D^{v}\langle
k,\omega(\xi)\rangle|\geq\beta_{0},$ (1.5)

where $\langle k,\omega\rangle=\sum\limits_{j=1}^{n}k_{j}\omega_{j}$,
$k\in\\{c=(c_{1},\dots,c_{n})\in\mathbb{R}^{n}:\lvert
c_{j}\rvert=1,j=1,\dots,n\\}$. We can take the smallest of such integer
$\bar{n}$. Moreover, by Rüssmann (Theorem 18.4 in [4]), we have

$\min_{\xi\in V}\max_{0\leq v\leq\bar{n}}\big{\lvert}D^{v}{\lvert
k\rvert_{2}}^{-2}\lvert\langle
k,\omega(\xi)\rangle\rvert^{2}\big{\rvert}\geq\beta_{0},$ (1.6)

for $\forall k\in\mathbb{Z}^{n}\setminus\\{0\\}$. As $\lvert\cdot\rvert$ and
$\lvert\cdot\rvert_{2}$ are equivalent, where $\lvert
k\rvert=\sum\limits_{j=1}^{n}\lvert k_{j}\rvert$, $\lvert
k\rvert_{2}=(\sum\limits_{j=1}^{n}\lvert k_{j}\rvert^{2})^{\frac{1}{2}}$, then
there exist $\bar{n}$ and $\beta$ such that

$\min_{\xi\in V}\max_{0\leq v\leq\bar{n}}|D^{v}\langle
k,\omega(\xi)\rangle|\geq\beta\lvert k\rvert,$ (1.7)

for $\forall k\in\mathbb{Z}^{n}\setminus\\{0\\}$, where
$\bar{n}=\bar{n}(\omega,V)\in\mathbb{N}$ is called the index of non-degeneracy
of $\omega$ with respect to $V$, and $\beta=\beta(\omega,V)>0$ is called the
amount of non-degeneracy of $\omega$ with respect to $V$.

Fixed $\tau\geq(n+2)(\bar{n}+1)$, $\gamma>0$ and $0<t<1$, introduce the
concept of Diophantine condition. The Diophantine condition refers to that
$\omega(\xi)$ satisfies

$\lvert e^{i\langle k,t\omega\rangle}-1\rvert\geq\frac{t\gamma}{{\lvert
k\rvert}^{\tau}},\ \forall\
k=(k_{1},k_{2},\dots,k_{n})\in\mathbb{Z}^{n}\setminus\\{0\\},$ (1.8)

where $\xi\in V_{\kappa}$, $\omega=(\omega_{1},\omega_{2},\dots,\omega_{n})$ ,
$\langle k,\omega\rangle=\sum\limits_{j=1}^{n}k_{j}\omega_{j}$ , $\lvert
k\rvert=\sum\limits_{j=1}^{n}\lvert k_{j}\rvert$. $\lvert\cdot\lvert_{r,s}$
denotes the supremum norm on $D_{r,s}$.

##  2 Main theorem

######  Theorem 2.1.

For the nearly integrable Hamiltonian system (1.1) , $H$ is real analytic on
$D_{r,s}\times V_{\kappa}$, and $H^{0}$is analytic on $V_{\kappa}$. If time
step $t$ and disturbance parameter $\epsilon$ are small enough, $\omega$
satisfies the Rüssmann’s non-degeneracy condition, then the generating
function representation of symplectic algorithm (1.2) has invariant tori on a
set of large measures. The generating function representation of phase flow in
nearly integrable Hamiltonian system (1.3) also has invariant tori on a set of
large measures, and there is only a slight deformation in the common area.
That is to say, for system (1.2), there is a non-empty Cantor set
$V_{\epsilon,t}\subseteq V$ and a Whitney smooth symplectic mapping
$\Phi_{\epsilon,t}:V_{\epsilon,t}\times\mathbb{T}^{n}\to\mathbb{R}^{n}\times\mathbb{T}^{n}$
that makes the following true.

  1. (i)

$\Phi_{\epsilon,t}$ is a symplectic conjugation between $G_{H^{\epsilon}}^{t}$
and $R_{\epsilon,t}$, i.e., $\Phi_{\epsilon,t}^{-1}\circ
G_{H^{\epsilon}}^{t}\circ\Phi_{\epsilon,t}=R_{\epsilon,t}$, where
$R_{\epsilon,t}$ is a rotation on $V_{\epsilon,t}\times\mathbb{T}^{n}$ with
frequency $t\omega_{\epsilon,t}$, i.e.,
$R_{\epsilon,t}(p,q)=(p,q+t\omega_{\epsilon,t})$.

  2. (ii)

$V_{\epsilon,t}$ is a set of positive measure if $\gamma$ is small enough. And
for $\gamma\to 0$, we have $\lvert V\setminus V_{\epsilon,t}\rvert\to 0$. To
be specific,

$\lvert V\setminus V_{\epsilon,t}\rvert\leq c_{7}\gamma
d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d),$ (2.1)

where $c_{7}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\hat{K}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$\hat{K}=\lvert\omega\rvert_{V}+c_{4}^{{}^{\prime}}(M_{1}+2nM_{2}l_{*})+1$,
$\mathcal{B}=(V\times(0,1)+\kappa)\cap\mathbb{R}^{n+1}$, $d$ is the diameter
of $V\times(0,1)$, $\bar{c}$ is a constant depending only on $n$, $\bar{n}$
and $\tau$, and $c_{4}^{{}^{\prime}}$ is a constant depending only on
$\tau,n,\bar{n},\gamma,\rho_{0},\sigma_{0}$ and $K_{0}$.

  3. (iii)

Let the time step be $t_{1}$ and $t_{2}$ respectively, and comppare the two
systems, then for $\xi\in V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}}$ we have

$\lvert\Phi_{\epsilon,t_{1}}-\Phi_{\epsilon,t_{2}}\rvert\leq
2c_{2}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),\
\lvert\omega_{\epsilon,t_{1}}-\omega_{\epsilon,t_{2}}\rvert\leq
2c_{4}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),$

where $M_{2}$ and $l_{*}$ are constants, and $c_{2}^{{}^{\prime}}$ and
$c_{4}^{{}^{\prime}}$ are constants depending only on
$\tau,n,\bar{n},\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$.
$V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}}$ is also a set of positive measure
if $\gamma$ is small enough. And for $\gamma\to 0$, we also have $\lvert
V\setminus(V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}})\rvert\to 0$.

##  3 Proof of Theorem 2.1

###  3.1 One-step Iterative Analysis

Firstly, we focus on one-step iteration. Set

$V_{\gamma,t,v}=\big{\\{}\xi\in V:\lvert e^{i\langle
k,t\omega_{v}(\xi)\rangle}-1\rvert\geq\frac{t\gamma_{v}}{{\lvert
k\rvert}^{\tau}},\ \forall\ k\in\mathbb{Z}^{n}\setminus\\{0\\}\big{\\}},$

that is, for $\xi\in V_{\gamma,t,v}$ , we have $\omega_{v}(\xi)$ satisfies the
Diophantine condition (1.8). It is a Cantor set, about which measure estimates
are given later. Whitney analytical extension was carried out for
$\omega_{v}(\xi)$ on $V_{\gamma,t,v}$ and denoted as
$V_{\kappa^{{}^{\prime}}_{v}}=V_{\gamma,t,v}+\kappa^{{}^{\prime}}_{v}$.

Denote the one-step transforming of the symplectic difference scheme as
$G_{v}:(p,q)\to(\hat{p},\hat{q})$ with

$G_{v}:\begin{cases}\hat{p}=p-\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)},\\\
\hat{q}=q+t\omega_{v}+\partial_{1}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)},\end{cases}$ (3.1)

here the subscript $v$ represents the $v$-th step. Assume the one-step
symplectic mapping can be expressed through the generating function $\psi_{v}$
as $\Psi_{v}:(I,\theta)\to(p,q)$ with

$\Psi_{v}:\begin{cases}I=p-\partial_{2}\psi_{v}(I,q),\\\
\theta=q+\partial_{1}\psi_{v},(I,q),\end{cases}$ (3.2)

so we have
$(I,\theta)\stackrel{{\scriptstyle\Psi_{v}}}{{\longrightarrow}}(p,q)\stackrel{{\scriptstyle
G_{v}}}{{\longrightarrow}}(\hat{p},\hat{q})\stackrel{{\scriptstyle\Psi_{v}^{-1}}}{{\longrightarrow}}(\hat{I},\hat{\theta})$,
and $\Psi_{v}^{-1}\circ G_{v}\circ\Psi_{v}$ can be expressed as follows:

$\displaystyle\hat{I}$
$\displaystyle=\hat{p}-\partial_{2}\psi_{v}(\hat{I},\hat{q})$
$\displaystyle=p-\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}-\partial_{2}\psi_{v}(\hat{I},\hat{q})$
$\displaystyle=I-\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}+\partial_{2}\psi_{v}(I,q)-\partial_{2}\psi_{v}(\hat{I},\hat{q}),$
$\displaystyle\hat{\theta}=$ $\displaystyle\
\hat{q}+\partial_{1}\psi_{v}(\hat{I},\hat{q})$ $\displaystyle=$
$\displaystyle\ q+t\omega_{v}+\partial_{1}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}+\partial_{1}\psi_{v}(\hat{I},\hat{q})$
$\displaystyle=$ $\displaystyle\
\theta+t\omega_{v}+\partial_{1}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}-\partial_{1}\psi_{v}(I,q)+\partial_{1}\psi_{v}(\hat{I},\hat{q}).$

Note that for $0<t<\delta$, we have

$\big{\lvert}\partial_{1}\big{(}t\epsilon
S_{0}(\hat{p},q)+t^{\alpha+1}P_{0}(\hat{p},q)\big{)}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq\frac{M_{1}t\epsilon+2nM_{2}l_{*}t^{\alpha+1}}{r/4},$
$\big{\lvert}\partial_{2}\big{(}t\epsilon
S_{0}(\hat{p},q)+t^{\alpha+1}P_{0}(\hat{p},q)]\big{)}\big{\rvert}_{\frac{r}{4},\frac{s}{4}}\leq\frac{M_{1}t\epsilon+2nM_{2}l_{*}t^{\alpha+1}}{s/4}.$

Thus, for $\epsilon$ and $t$ small enough such that
$\frac{M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}}{r/4},\frac{M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}}{s/4}\leq
E_{0}$, then, if $r_{0}\leq\frac{r}{4}$, $s_{0}\leq\frac{s}{4}$, we can denote
$\big{\lvert}\partial_{1}(t\epsilon
S_{0}+t^{\alpha+1}P_{0})\big{\rvert}_{r_{0},s_{0}}\leq tE_{0}$,
$\big{\lvert}\partial_{2}(t\epsilon
S_{0}+t^{\alpha+1}P_{0})\big{\lvert}_{r_{0},s_{0}}\leq tE_{0}$ , here $E_{0}$
is also small enough as $\epsilon$ and $t$ are small enough.

Similarly, we can assume that $\big{\lvert}\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{\rvert}_{r_{v},s_{v}}\leq tE_{v}$ as well as
$\big{\lvert}\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{\lvert}_{r_{v},s_{v}}\leq tE_{v}$, where $E_{v}$
is also a small quantity. Then, as $\psi_{v}$ and $(t\epsilon
S_{v}+t^{\alpha+1}P_{v})$ are small quantities, we can get the following
analysis:

$\displaystyle\partial_{1}\psi_{v}(I,q)=\ $
$\displaystyle\partial_{1}\psi_{v}\big{(}I,\theta-\partial_{1}\psi_{v}(I,q)\big{)}=\partial_{1}\psi_{v}(I,\theta)+o(E_{v}),$
$\displaystyle\partial_{1}\psi_{v}(\hat{I},\hat{q})=\ $
$\displaystyle\partial_{1}\psi_{v}\big{(}I-\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)+\partial_{2}\psi_{v}(I,q)-\partial_{2}\psi_{v}(\hat{I},\hat{q}),$
$\displaystyle\theta+t\omega_{v}+\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)-\partial_{1}\psi_{v}(I,q)\big{)}$
$\displaystyle=\ $
$\displaystyle\partial_{1}\psi_{v}\big{(}I,\theta+t\omega_{v}(\xi)\big{)}+o(E_{v}),$
$\displaystyle\partial_{1}(t\epsilon S_{v}+t^{\alpha+1}$ $\displaystyle
P_{v})(\hat{p},q)=\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{(}I-\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)$ $\displaystyle\
+\partial_{2}\psi_{v}(I,q),\theta-\partial_{1}\psi_{v}(I,q)\big{)}$
$\displaystyle=\ $ $\displaystyle\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(I,\theta)+o(E_{v}).$

Similarly, we have

$\displaystyle\partial_{2}\psi_{v}(I,q)=\ $
$\displaystyle\partial_{2}\psi_{v}\big{(}I,\theta-\partial_{1}\psi_{v}(I,q)\big{)}=\partial_{2}\psi_{v}(I,\theta)+o(E_{v}),$
$\displaystyle\partial_{2}\psi_{v}(\hat{I},\hat{q})=\ $
$\displaystyle\partial_{2}\psi_{v}\big{(}I-\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)+\partial_{2}\psi_{v}(I,q)-\partial_{2}\psi_{v}(\hat{I},\hat{q}),$
$\displaystyle\theta+t\omega_{v}+\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)-\partial_{1}\psi_{v}(I,q)\big{)}$
$\displaystyle=\ $
$\displaystyle\partial_{2}\psi_{v}\big{(}I,\theta+t\omega_{v}(\xi)\big{)}+o(E_{v}),$
$\displaystyle\partial_{2}(t\epsilon S_{v}+t^{\alpha+1}$ $\displaystyle
P_{v})(\hat{p},q)=\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{(}I-\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(\hat{p},q)$ $\displaystyle\
+\partial_{2}\psi_{v}(I,q),\theta-\partial_{1}\psi_{v}(I,q)\big{)}$
$\displaystyle=\ $ $\displaystyle\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})(I,\theta)+o(E_{v}).$

So, we have $\Psi_{v}^{-1}\circ G_{v}\circ\Psi_{v}$ can be expressed as
follows:

$\displaystyle\hat{I}=$ $\displaystyle\ I-\partial_{2}\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}+\partial_{2}\psi_{v}(I,\theta)$
$\displaystyle-\partial_{2}\psi_{v}\big{(}I,\theta+t\omega_{v}\big{)}+o(E_{v}),$
$\displaystyle\hat{\theta}=$ $\displaystyle\
\theta+t\omega_{v}(\xi)+\partial_{1}\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}-\partial_{1}\psi_{v}(I,\theta)$
$\displaystyle+\partial_{1}\psi_{v}\big{(}I,\theta+t\omega_{v}\big{)}+o(E_{v}).$

Looking at this expression, we can truncate $t\epsilon
S_{v}+t^{\alpha+1}P_{v}$ by zero degree term and one degree term with respect
to the action variables to get $(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}$. For
general function $f(I,\theta)$, entry notation $[f]$ and Fourier coefficients
$f_{k}(I)$,

$\displaystyle f(I,\theta)=\sum\limits_{k\in\mathbb{Z}^{n}}f_{k}(I)e^{i\langle
k,\theta\rangle}=\widetilde{f(I,\theta)}+[f],$
$\displaystyle\widetilde{f(I,\theta)}=\sum\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}f_{k}(I)e^{i\langle
k,\theta\rangle}.$

To get $\psi_{v}$, we consider the following equation:

$\psi_{v}\big{(}I,\theta+t\omega_{v}\big{)}-\psi_{v}(I,\theta)+\widetilde{\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}}=0,$

where

$\displaystyle\ \widetilde{\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}}$ $\displaystyle=$
$\displaystyle\ \big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}-\big{[}\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}\big{]}$
$\displaystyle=$
$\displaystyle\sum\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}(I)e^{i\langle k,\theta\rangle}.$

Here $\big{[}(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}\big{]}$ is the mean
values of $(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}$ with respect to the angle
variables on $\mathbb{T}^{n}$, i.e.

$\displaystyle\big{[}(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}\big{]}$
$\displaystyle=\frac{1}{(2\pi)^{n}}\int_{\mathbb{T}^{n}}\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}d\theta.$

Define $\omega_{v+1}=\omega_{v}+\big{[}\partial_{1}(\epsilon
S_{v}+t^{\alpha}P_{v})^{*}\big{(}\xi)]$. By the way, $[\partial_{2}(\epsilon
S_{v}+t^{\alpha}P_{v})^{*}\big{]}=0$. Moreover, we truncate the Fourier series
expansion of $(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}$ with respect to angle
variables by order $K_{v}$. And then we get the final homological equation:

$\psi_{v}(I,\theta+t\omega_{v})-\psi_{v}(I,\theta)+T_{K_{v}}\widetilde{\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}}=0,$ (3.3)

where

$\displaystyle T_{K_{v}}\widetilde{\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}}=\sum\limits_{0<\lvert
k\rvert\leq K_{v}}\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle k,\theta\rangle}.$

To solve $\psi_{v}$ in the homological equation (3.3), we can expand
$\psi_{v}(I,\theta+t\omega_{v})$, $\psi_{v}(I,\theta)$ and $\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}(I,\theta)\big{)}^{*}$ by Fourier series, and
then compare their coefficients. Their Fourier series expansion are shown
below:

$\displaystyle\psi_{v}(I,\theta+t\omega_{v})=$
$\displaystyle[\psi_{v}](I)+\sum\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\big{(}\psi_{v}\big{)}_{k}e^{i\langle
k,\theta+t\omega_{v}\rangle},$ $\displaystyle\psi_{v}(I,\theta)=[\psi_{v}](I)$
$\displaystyle+\sum\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\big{(}\psi_{v}\big{)}_{k}e^{i\langle
k,\theta\rangle},$ $\displaystyle\big{(}t\epsilon
S_{v}(I,\theta)+t^{\alpha+1}P_{v}$
$\displaystyle(I,\theta)\big{)}^{*}=\big{[}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{]}(I)$
$\displaystyle+\sum\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle k,\theta\rangle}.$

Compare their coefficients of $e^{i\langle k,\theta\rangle}$, then we have
$\big{(}\psi_{v}\big{)}_{k}=\frac{-\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}}{e^{i\langle
k,t\omega_{v}\rangle}-1}$, so we get $\psi_{v}$ :

$\psi_{v}(I,\theta)=-\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\frac{\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle
k,\theta\rangle}}{e^{i\langle k,t\omega_{v}\rangle}-1}.$ (3.4)

######  Lemma 3.1.

(see Lemma A.1 in [3]) If $f\in A^{s}$, then
$f=\sum\limits_{k}f_{k}e^{i\langle k,\theta\rangle}$ with

$\lvert f_{k}\rvert\leq\lvert f\rvert_{s}e^{-\lvert k\rvert s},\
k\in\mathbb{Z}^{n}$

where $A^{s}$ is the set of all real analytic functions on
$\\{\theta:\lvert\Im(\theta)\rvert<s\\}\subset\mathbb{C}^{n}$ with sup-norm
$\lvert\cdot\rvert_{s}$, $\lvert k\rvert=\lvert k_{1}\rvert+\dots+\lvert
k_{n}\rvert$.

By Lemma 3.1, $\lvert\big{(}\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}\rvert\leq\lvert\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\rvert_{s_{v}}e^{-\lvert k\rvert s_{v}}$.
$\omega_{v}$ satisfies the Diophantine condition (1.8) on $V_{\gamma,t,v}$,
and we have

$\displaystyle\lvert\partial_{1}\psi_{v}\rvert_{s_{v}-\sigma_{v}}$
$\displaystyle\leq\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\frac{\lvert\partial_{1}\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle
k,\theta\rangle}\rvert_{{s_{v}-\sigma_{v}}}}{\lvert e^{i\langle
k,t\omega_{v}\rangle}-1\rvert}$ $\displaystyle\leq\sum\limits_{0<\lvert
k\rvert\leq K_{v}}\frac{\lvert\big{(}\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}\rvert e^{\lvert
k\rvert(s_{v}-\sigma_{v})}}{\lvert e^{i\langle k,t\omega_{v}\rangle}-1\rvert}$
$\displaystyle\leq\frac{1}{t\gamma_{v}}\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\lvert\big{(}\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}\rvert e^{\lvert
k\rvert(s_{v}-\sigma_{v})}\lvert k\rvert^{\tau}$
$\displaystyle\leq\frac{\lvert\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\rvert_{s_{v}}}{t\gamma_{v}}\sum\limits_{0<\lvert
k\rvert\leq K_{v}}e^{-\lvert k\rvert\sigma_{v}}\lvert k\rvert^{\tau}$
$\displaystyle\leq\frac{c_{1}}{t\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\rvert_{s_{v}}$
$\displaystyle\leq\frac{2c_{1}}{t\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\rvert_{s_{v}},$

where $c_{1}$ is a constant only depends on $\tau$ and $n$. Similarly, we have

$\displaystyle\lvert\partial_{2}\psi_{v}\rvert_{s_{v}-\sigma_{v}}$
$\displaystyle\leq\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\frac{\lvert\partial_{2}\big{(}((t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*})_{k}e^{i\langle
k,\theta\rangle}\big{)}\rvert_{{s_{v}-\sigma_{v}}}}{\lvert e^{i\langle
k,t\omega_{v}\rangle}-1\rvert}$ $\displaystyle\leq\sum\limits_{0<\lvert
k\rvert\leq K_{v}}\frac{\lvert ki\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle
k,\theta\rangle}\rvert_{{s_{v}-\sigma_{v}}}}{\lvert e^{i\langle
k,t\omega_{v}\rangle}-1\rvert}$ $\displaystyle\leq\sum\limits_{0<\lvert
k\rvert\leq K_{v}}\frac{\lvert ki\big{(}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}\rvert e^{\lvert
k\rvert(s_{v}-\sigma_{v})}}{\lvert e^{i\langle k,t\omega_{v}\rangle}-1\rvert}$
$\displaystyle\leq\frac{1}{t\gamma_{v}}\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\lvert ki\big{(}(t\epsilon S_{v}+t^{\alpha+1}P_{v})^{*}\big{)}_{k}\rvert
e^{\lvert k\rvert(s_{v}-\sigma_{v})}\lvert k\rvert^{\tau}$
$\displaystyle\leq\frac{\lvert\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\rvert_{s_{v}}}{t\gamma_{v}}\sum\limits_{0<\lvert
k\rvert\leq K_{v}}e^{-\lvert k\rvert\sigma_{v}}\lvert k\rvert^{\tau}$
$\displaystyle\leq\frac{c_{1}}{t\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}\rvert_{s_{v}}$
$\displaystyle\leq\frac{2c_{1}}{t\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\rvert_{s_{v}}.$

Now we give the immature parameter settings, and the final complete parameter
settings will come later.

$\begin{array}[]{ccc}\tau_{v}=\frac{1^{-2}+\dots+v^{-2}}{2\sum\limits_{v=1}^{\infty}v^{-2}},&s_{v}=\frac{1}{4}(1-\tau_{v})s_{0},&\sigma_{v}=\frac{1}{4}(s_{v}-s_{v+1}),\\\
K_{v+1}=4K_{v},&r_{v+1}=\eta_{v}r_{v},&\rho_{v}=\frac{1}{4}(r_{v}-r_{v+1}),\\\
\gamma_{v}=\frac{\gamma^{\bar{n}+1}}{2^{(\bar{n}+1)v}},&F_{v}=\frac{E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n+1}\rho_{v}},&F_{v+1}=F_{v}^{\frac{v+1}{v}},\\\
\eta_{v}=\frac{1}{12c_{3}}F_{v}^{\frac{1}{v}}.\end{array}$

Note: $c_{3}$ is a constant. In fact, $\eta_{v}$ is also a constant, as will
be explained later.

Denote $\Phi_{v}=\Psi_{0}\circ\Psi_{1}\circ\dots\circ\Psi_{v-1}$, and let
$C_{v}=R_{v}^{-1}\circ G_{v}$, where $R_{v}$ is a rotation with frequency
$t\omega_{v}$, i.e., $R_{v}(I,\theta)=(I,\theta+t\omega_{v})$. Focus on one-
step iteration, we have the following iteration lemma.

######  Lemma 3.2.

(Iteration lemma)

Assume that $\big{\lvert}\partial_{1}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{\rvert}_{r_{v},s_{v}},\big{\lvert}\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\big{\rvert}_{r_{v},s_{v}}\leq tE_{v}$, $t\omega_{v}$
satisfies the Diophantine condition (1.8) with $t\gamma_{v}$, and other
parameters are the same as above. Take the appropriate initial values, then
for $v\geq 3$, there exist a real analytic symplectic mapping $\Phi_{v}$ which
is defined on $D_{\eta_{v}r_{v},s_{v}-4\sigma_{v}}\times
V_{\kappa^{{}^{\prime}}_{v}}$ such that $\Phi_{v}^{-1}\circ
G_{H^{\epsilon}}^{t}\circ\Phi_{v}:(p,q)\to(\hat{p},\hat{q})$

$\begin{cases}\hat{p}=p-\partial_{2}\big{(}t\epsilon
S_{v+1}(\hat{p},q)+t^{\alpha+1}P_{v+1}(\hat{p},q)\big{)},\\\
\hat{q}=q+t\omega_{v+1}+\partial_{1}\big{(}t\epsilon
S_{v+1}(\hat{p},q)+t^{\alpha+1}P_{v+1}(\hat{p},q)\big{)},\end{cases}$

where $\big{\lvert}\partial_{1}(t\epsilon
S_{v+1}+t^{\alpha+1}P_{v+1})\big{\rvert}_{r_{v+1},s_{v+1}},\big{\lvert}\partial_{2}(t\epsilon
S_{v+1}+t^{\alpha+1}P_{v+1})\big{\rvert}_{r_{v+1},s_{v+1}}\leq tE_{v+1}$, and
the following results are true.

  1. (i)

$\Phi_{v}$ is well defined and real analytic on
$D_{r_{v-1}-3\rho_{v-1},s_{v-1}-3\sigma_{v-1}}\times
V_{\kappa^{{}^{\prime}}_{v}}$, and it has the following estimate:

$\lvert\Phi_{v}-\Phi_{v-1}\rvert\leq\frac{4c_{1}E_{v-1}}{\gamma_{v-1}\sigma_{v-1}^{\tau+n}},$
(3.5)

where $c_{1}$ is a constant only depends on $\tau$ and $n$.

  2. (ii)

$C_{v}=R_{v}^{-1}\circ\Phi_{v}^{-1}\circ G_{H^{\epsilon}}^{t}\circ\Phi_{v}$ is
well defined and real analytic on $D_{r_{v},s_{v}}\times
V_{\kappa^{{}^{\prime}}_{v}}$, and it has the following estimate:

$\lvert C_{v}-id\rvert_{r_{v},s_{v}}\leq tE_{v}.$ (3.6)

  3. (iii)

For $\omega=\omega_{0}$, on $V_{\kappa^{{}^{\prime}}_{v}}$ we have

$\lvert\omega_{v}-\omega_{v-1}\rvert\leq 2E_{v}.$ (3.7)

###  3.2 Proof of the Iteration Lemma

Firstly, $C_{v}=R_{v}^{-1}\circ G_{v}$ :

$\begin{cases}\hat{p}=p-\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)},\\\
\hat{q}=q+\partial_{1}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}.\end{cases}$

In order to express $C_{v}$ explicitly, firstly, let us solve the first
equation:

$\hat{p}=p-\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}.$

Let $X_{1}$ be the set of all real analytic functions $p_{*}(p,q)$ on
$D_{r_{v},s_{v}}$, where $p_{*}(p,q)$ is periodic $2\pi$ with respect to angle
variables, and $\lvert p_{*}(p,q)\rvert_{r_{v},s_{v}}\leq\rho_{v}$.

Consider mapping:

$f_{1}:p_{*}\to f_{1}(p_{*})=-\partial_{2}\big{(}t\epsilon
S_{v}(p+p_{*},q)+t^{\alpha+1}P_{v}(p+p_{*},q)\big{)}.$

We know that $f_{1}$ is well defined and real analytic on $X_{1}$, and
$f_{1}:X_{1}\to X_{1}$. By Cauchy estimates, for $\forall\ p_{*}\in X_{1}$,
$(p,q)\in D_{r_{v},s_{v}}$we have

$\lvert f_{1}(p_{*})\rvert\leq\lvert\partial_{2}\big{(}t\epsilon
S_{v}(p+p_{*},q)+t^{\alpha+1}P_{v}(p+p_{*},q)\big{)}\rvert\leq\rho_{v}.$

For $\forall\ p_{*}^{1},p_{*}^{2}\in X_{1}$, $(p,q)\in D_{r_{v},s_{v}}$, we
have

$\displaystyle\lvert f_{1}(p_{*}^{1})-f_{1}(p_{*}^{2})\rvert=$ $\displaystyle\
\lvert\partial_{2}\big{(}t\epsilon
S_{v}(p+p_{*}^{1},q)+t^{\alpha+1}P_{v}(p+p_{*}^{1},q)\big{)}$
$\displaystyle-\partial_{2}\big{(}t\epsilon
S_{v}(p+p_{*}^{2},q)+t^{\alpha+1}P_{v}(p+p_{*}^{2},q)\big{)}\rvert$
$\displaystyle\leq$ $\displaystyle\
n\lvert\partial_{1}\partial_{2}\big{(}t\epsilon
S_{v}(p+\bar{p},q)+t^{\alpha+1}P_{v}(p+\bar{p},q)\big{)}\rvert\cdot\lvert
p_{*}^{1}-p_{*}^{2}\rvert$ $\displaystyle\leq$ $\displaystyle\
\frac{ntE_{v}}{\rho_{v}}\cdot\lvert p_{*}^{1}-p_{*}^{2}\rvert$
$\displaystyle\leq$ $\displaystyle\ L_{1}\lvert p_{*}^{1}-p_{*}^{2}\rvert,$

if $\frac{ntE_{0}}{\rho_{0}}<1$, where $0<L_{1}<1,\bar{p}\in X_{1}$.

So, $f_{1}:X_{1}\to X_{1}$ is a compressed mapping, by the fixed point
theorem, there is a unique fixed point $p_{**}\in X_{1}$, such that $C_{v}$
can be expressed explicitly as:

$\begin{cases}\hat{p}=p+p_{**},\\\ \hat{q}=q+\partial_{1}\big{(}t\epsilon
S_{v}(p+p_{**},q)+t^{\alpha+1}P_{v}(p+p_{**},q)\big{)},\end{cases}$

so we get the following corollary.

######  Corollary 3.1.

$\partial_{1}(t\epsilon S_{v}+t^{\alpha+1}P_{v})(p+p_{**},q,\xi)$ ,
$\partial_{2}(t\epsilon S_{v}+t^{\alpha+1}P_{v})(p+p_{**},q,\xi)$ are well
defined and real analytic on $D_{1}=D_{r_{v}-\rho_{v},s_{v}-\sigma_{v}}\times
V_{\kappa^{{}^{\prime}}_{v}}$, and they are periodic $2\pi$ with respect to
angle variables. Moreover, there are the following estimates on $D_{1}$:

$\lvert\partial_{1}(t\epsilon S_{v}+t^{\alpha+1}P_{v})\rvert_{D_{1}}\leq
tE_{v},\ \lvert\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\rvert_{D_{1}}\leq tE_{v}.$

Similarly, let us use the fixed point theorem again to express $\Psi_{v}$
explicitly. Consider the second equation in (3.2):

$\theta=q+\partial_{1}\psi_{v}(I,q).$

Let $X_{2}$ be the set of all real analytic functions $\theta_{*}(I,\theta)$
on $D_{r_{v},s_{v}}$, where $\theta_{*}(I,\theta)$ is periodic $2\pi$ with
respect to angle variables,
and$\lvert\theta_{*}(I,\theta)\rvert_{r_{v},s_{v}}\leq\sigma_{v}$. Consider
mapping:

$f_{2}:\theta_{*}\to
f_{2}(\theta_{*})=\partial_{1}\psi_{v}(I,\theta-\theta_{*}),$

We know that $f_{2}$ is well defined and real analytic on $X_{2}$, and
$f_{2}:X_{2}\to X_{2}$. By Cauchy estimates, for $\forall\ \theta_{*}\in
X_{2},(I,\theta)\in D_{r_{v},s_{v}}$, we have

$\lvert
f_{2}(\theta_{*})\rvert\leq\lvert\partial_{1}\psi_{v}(I,\theta-\theta_{*})\rvert\leq\sigma_{v}.$

For $\forall\ \theta_{*}^{1},\theta_{*}^{2}\in X_{2}$, $(I,\theta)\in
D_{r_{v},s_{v}}$, we have

$\displaystyle\lvert f_{2}(\theta_{*}^{1})-f_{2}(\theta_{*}^{2})\rvert=$
$\displaystyle\
\lvert\partial_{1}\psi_{v}(I,\theta-\theta_{*}^{1})-\partial_{1}\psi_{v}(I,\theta-\theta_{*}^{2})\rvert$
$\displaystyle\leq$ $\displaystyle\
\frac{2nc_{1}\lvert\partial_{1}\partial_{2}\big{(}t\epsilon
S_{v}(I,\theta-\bar{\theta})+t^{\alpha+1}P_{v}(I,\theta-\bar{\theta})\big{)}\rvert}{t\gamma_{v}\sigma_{v}^{\tau+n}}\cdot\lvert
p_{*}^{1}-p_{*}^{2}\rvert$ $\displaystyle\leq$ $\displaystyle\
\frac{2nc_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}\sigma_{v}}\cdot\lvert
p_{*}^{1}-p_{*}^{2}\rvert$ $\displaystyle\leq$ $\displaystyle\ L_{2}\lvert
p_{*}^{1}-p_{*}^{2}\rvert,$

if $\frac{2nc_{1}E_{0}}{\gamma_{0}\sigma_{0}^{\tau+n+1}}<1$, where
$0<L_{2}<1,\bar{\theta}\in X_{2}$.

So, $f_{2}:X_{2}\to X_{2}$ is a compressed mapping, by the fixed point
theorem, there is a unique fixed point $\theta_{**}\in X_{2}$, such that
$\Psi_{v}$ can be expressed explicitly as:

$\begin{cases}p=I+\partial_{2}\psi_{v}(I,\theta-\theta_{**}),\\\
q=\theta-\theta_{**},\end{cases}$

so we get the following corollaries.

######  Corollary 3.2.

$\partial_{1}\psi_{v}(I,\theta-\theta_{**},\xi)$ and
$\partial_{2}\psi_{v}(I,\theta-\theta_{**},\xi)$ are well defined and real
analytic on $D_{2}$, where $D_{2}=D_{r_{v}-2\rho_{v},s_{v}-2\sigma_{v}}\times
V_{\kappa^{{}^{\prime}}_{v}}$, and it is periodic $2\pi$ with respect to angle
variables. Moreover, there are the following estimates on $D_{2}$:

$\lvert\partial_{1}\psi_{v}\rvert_{D_{2}}\leq\frac{2c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}},\
\lvert\partial_{2}\psi_{v}\rvert_{D_{2}}\leq\frac{2c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}}.$

######  Corollary 3.3.

$\Psi_{v}$ and $\Psi_{v}^{-1}$ are well defined and real analytic on $D_{3}$,
where $D_{3}=D_{r_{v}-3\rho_{v},s_{v}-3\sigma_{v}}\times
V_{\kappa^{{}^{\prime}}_{v}}$, and they are both from $D_{3}$ to $D_{2}$.
Meanwhile, there is the following estimate on $D_{3}$:

$\lvert\Psi_{v}-id\rvert_{D_{3}}\leq\frac{2c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}},$

so $\Psi_{v}^{-1}\circ R_{v}\circ C_{v}\circ\Psi_{v}$ is well defined on
$D_{r_{v+1},s_{v+1}}\times V_{\kappa^{{}^{\prime}}_{v}}$, and from
$D_{r_{v+1},s_{v+1}}\times V_{\kappa^{{}^{\prime}}_{v}}$ to $D_{1}$.

Let $\Phi_{v+1}=\Phi_{v}\circ\Psi_{v}$, as we know

$\displaystyle D\Phi_{v}$
$\displaystyle=D(\Psi_{0}\circ\Psi_{1}\circ\dots\circ\Psi_{v-1})$
$\displaystyle\leq\lvert D\Psi_{0}\rvert\lvert D\Psi_{1}\rvert\dots\lvert
D\Psi_{v-1}\rvert$
$\displaystyle\leq\prod\limits_{v}(1+\frac{2c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}})$
$\displaystyle\leq 2,$

if $\frac{2c_{1}E_{0}}{\gamma_{0}\sigma_{0}^{\tau+n}}\leq 1$. Then, on
$D_{r_{v}-3\rho_{v},s_{v}-3\sigma_{v}}$, we have

$\displaystyle\lvert\Phi_{v+1}-\Phi_{v}\rvert$
$\displaystyle=\lvert\Phi_{v}\circ\Psi_{v}-\Phi_{v}\rvert$
$\displaystyle\leq\lvert D\Phi_{v}\rvert\cdot\lvert\Psi_{v}-I\rvert$
$\displaystyle\leq\frac{4c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}},$

which means that $(i)$ in Lemma 3.2 (Iteration Lemma) holds. Besides,

$\displaystyle\lvert\omega_{v+1}-\omega_{v}\rvert$
$\displaystyle=\big{\lvert}\big{[}\partial_{1}(\epsilon
S_{v}+t^{\alpha}P_{v})^{*}\big{]}\big{\rvert}_{r_{v+1},s_{v+1}}\leq 2E_{v},$

which means that $(iii)$ in Lemma 3.2 (Iteration Lemma) holds.

Next, we proof $(ii)$ in Lemma 3.2 (Iteration Lemma). Let
$C_{v+1}=R_{v+1}^{-1}\circ\Psi_{v}^{-1}\circ R_{v}\circ C_{v}\circ\Psi_{v}$,
$C_{v+1}$ is well defined and real analytic on $D_{r_{v+1},s_{v+1}}$, and then
$C_{v+1}:(I,\theta)\to(\hat{I},\hat{\theta})$ can be expressed as follows:

$\begin{cases}\hat{I}=I+\partial_{2}\psi_{v}(I,q)-\partial_{2}\psi_{v}(\hat{I},\hat{q})-\partial_{2}\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}(\hat{p},q),\\\
\hat{\theta}=\theta-\partial_{1}\psi_{v}(I,q)+\partial_{1}\psi_{v}(\hat{I},\hat{q})+\partial_{1}\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}(\hat{p},q)-t\omega_{v+1}+t\omega_{v}.\end{cases}$

Then, on $D_{r_{v+1},s_{v+1}}$, we have

$\displaystyle\lvert\hat{I}-I\rvert=\ $
$\displaystyle\lvert\partial_{2}\psi_{v}(\hat{I},\hat{q})-\partial_{2}\psi_{v}(I,q)+\partial_{2}\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}(\hat{p},q)\rvert$ $\displaystyle\leq\ $
$\displaystyle I_{1}+I_{2}+\dots+I_{5},$
$\displaystyle\lvert\hat{\theta}-\theta\rvert=\ $
$\displaystyle\lvert\partial_{1}\psi_{v}(\hat{I},\hat{q})-\partial_{1}\psi_{v}(I,q)+\partial_{1}\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}(\hat{p},q)-t\omega_{v+1}+t\omega_{v}\rvert$
$\displaystyle\leq\ $ $\displaystyle J_{1}+J_{2}+\dots+J_{5},$

where

$\displaystyle I_{1}$
$\displaystyle=\lvert\partial_{2}\psi_{v}(\hat{I},\hat{q})-\partial_{2}\psi_{v}(I,\hat{q})\rvert,$
$\displaystyle I_{2}$
$\displaystyle=\lvert\partial_{2}\psi_{v}(I,\hat{q})-\partial_{2}\psi_{v}(I,q+t\omega_{v})\rvert,$
$\displaystyle I_{3}$
$\displaystyle=\lvert\partial_{2}\psi_{v}(I,q+t\omega_{v})-\partial_{2}\psi_{v}(I,q)+\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(I,q)+t^{\alpha+1}P_{v}(I,q)\big{)}^{*}}\rvert,$ $\displaystyle I_{4}$
$\displaystyle=\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}^{*}}-\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(I,q)+t^{\alpha+1}P_{v}(I,q)\big{)}^{*}}\rvert,$ $\displaystyle I_{5}$
$\displaystyle=\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}}-\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}^{*}}\rvert,$
$\displaystyle J_{1}$
$\displaystyle=\lvert\partial_{1}\psi_{v}(\hat{I},\hat{q})-\partial_{1}\psi_{v}(I,\hat{q})\rvert,$
$\displaystyle J_{2}$
$\displaystyle=\lvert\partial_{1}\psi_{v}(I,\hat{q})-\partial_{1}\psi_{v}(I,q+t\omega_{v})\rvert,$
$\displaystyle J_{3}$
$\displaystyle=\lvert\partial_{1}\psi_{v}(I,q+t\omega_{v})-\partial_{1}\psi_{v}(I,q)+\partial_{1}\widetilde{\big{(}t\epsilon
S_{v}(I,q)+t^{\alpha+1}P_{v}(I,q)\big{)}^{*}}\rvert,$ $\displaystyle J_{4}$
$\displaystyle=\lvert\partial_{1}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}^{*}}-\partial_{1}\widetilde{\big{(}t\epsilon
S_{v}(I,q)+t^{\alpha+1}P_{v}(I,q)\big{)}^{*}}\rvert,$ $\displaystyle J_{5}$
$\displaystyle=\lvert\partial_{1}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}}-\partial_{1}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}^{*}}\rvert.$

######  Lemma 3.3.

(Lemma A.2 in [3]) If $f\in A^{s}$, then

$\lvert f-T_{K}f\rvert_{s-\sigma}\leq cK^{n}e^{-K\sigma}\lvert f\rvert_{s},\
0\leq\sigma\leq s,$

where the constant $c$ only depends on $n$, $A^{s}$ is in Lemma 3.1.

Next, by Lemma 3.1, Lemma 3.3 and the Cauchy estimates, we estimate
$I_{1},I_{2},\dots,I_{5}$:

$\displaystyle I_{1}$
$\displaystyle=\lvert\partial_{2}\psi_{v}(\hat{I},\hat{q})-\partial_{2}\psi_{v}(I,\hat{q})\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq\frac{n\lvert\partial_{2}\psi_{v}\rvert_{r_{v}-3\rho_{v},s_{v}-4\sigma_{v}}}{\rho_{v}}\lvert\hat{I}-I\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq\frac{2nc_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\lvert\hat{I}-I\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}.$
$\displaystyle I_{2}=$ $\displaystyle\
\lvert\partial_{2}\psi_{v}(I,\hat{q})-\partial_{2}\psi_{v}(I,q+t\omega_{v})\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{n\lvert\partial_{2}\psi_{v}\rvert_{r_{v}-4\rho_{v},s_{v}-3\sigma_{v}}}{\sigma_{v}}\lvert\hat{q}-q-t\omega_{v}\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{2nc_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}\sigma_{v}}\cdot\lvert\hat{q}-q-t\omega_{v}\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{2nc_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}\sigma_{v}}\cdot\lvert\partial_{1}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{2nc_{1}tE_{v}^{2}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}.$ $\displaystyle
I_{3}=$ $\displaystyle\
\lvert\partial_{2}\psi_{v}(I,q+t\omega_{v})-\partial_{2}\psi_{v}(I,q)+\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}}(I,q)\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}}(I,q)-T_{K_{v}}\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}}(I,q)\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(I,q)+t^{\alpha+1}P_{v}(I,q)\big{)}^{*}}\rvert_{r_{v}-4\rho_{v},s_{v}-3\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}tE_{v},$

just like Lemma 3.3, here $c_{2}$ is a constant only depends on $n$.

$\displaystyle I_{4}=$ $\displaystyle\
\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}}(\hat{p},q)-\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}}(I,q)\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{n\lvert\partial_{2}\widetilde{(t\epsilon
S_{v}+t^{\alpha+1}P_{v})^{*}}\rvert_{r_{v}-3\rho_{v},s_{v}-4\sigma_{v}}}{\rho_{v}}\lvert\hat{p}-I\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{4ntE_{v}}{\rho_{v}}\\{\lvert\big{(}\partial_{2}\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}\rvert+\lvert\partial_{2}\psi_{v}(I,q)\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}\\}$
$\displaystyle\leq$ $\displaystyle\
\frac{4ntE_{v}}{\rho_{v}}(tE_{v}+\frac{2c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}})$
$\displaystyle\leq$ $\displaystyle\
\frac{8nc_{1}tE_{v}^{2}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}.$
$\displaystyle I_{5}=$ $\displaystyle\
\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}}(\hat{p},q)-\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}+t^{\alpha+1}P_{v}\big{)}^{*}(\hat{p},q)}\rvert_{r_{v}-4\rho_{v},s_{v}-4\sigma_{v}}$
$\displaystyle\leq$ $\displaystyle\
\frac{\eta_{v}^{2}}{1-\eta_{v}}\lvert\partial_{2}\widetilde{\big{(}t\epsilon
S_{v}(\hat{p},q)+t^{\alpha+1}P_{v}(\hat{p},q)\big{)}}\rvert_{r_{v},s_{v}}$
$\displaystyle\leq$ $\displaystyle\ 4\eta_{v}^{2}tE_{v},$

if $\eta_{v}<\frac{1}{2}$.

In conclusion, if
$\frac{2nc_{1}E_{0}}{\gamma_{0}\sigma_{0}^{\tau+n}\rho_{0}}<1$, then we have

$\displaystyle\ \lvert\hat{I}-I\rvert_{r_{v+1},s_{v+1}}$ $\displaystyle\leq$
$\displaystyle\ \big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}.$

Similarly, we can estimate $J_{1},J_{2},\dots,J_{5}$ the same way, then we
have

$\displaystyle\ \lvert\hat{\theta}-\theta\rvert_{r_{v+1},s_{v+1}}$
$\displaystyle\leq$ $\displaystyle\ \big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}.$

For $F_{v}=\frac{E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n+1}\rho_{v}}$, if
$\epsilon$ and $t$ are small enough, we have $F_{0}$ is small enough such that
$2nc_{1}F_{v}\leq L_{0},(0<L_{0}<1)$, then

$\displaystyle\ \frac{\big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}}{\gamma_{v+1}\sigma_{v+1}^{\tau+n+1}\rho_{v+1}}$ $\displaystyle\leq$
$\displaystyle\ \frac{c_{3}\big{\\{}\big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}\big{\\}}}{\gamma_{v}\sigma_{v}^{\tau+n+1}\eta_{v}\rho_{v}}$
$\displaystyle\leq$ $\displaystyle\
(10nc_{1}c_{3})\cdot\frac{tF_{v}^{2}}{\eta_{v}}+(4c_{2}c_{3}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4c_{3}\eta_{v}^{2})\cdot\frac{tF_{v}}{\eta_{v}},$

where $c_{3}=2^{\tau+\bar{n}+n+2}$. We let
$\eta_{v}=\frac{1}{12c_{3}}F_{v}^{\frac{1}{v}}$ for $v\geq 3$. If $K_{0}$ is
big enough such that $144c_{2}c_{3}^{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}\leq
F_{v}^{\frac{2}{v}}$, and if $F_{0}$ is small enough such that
$360nc_{1}c_{3}^{2}\leq F_{v}^{-(1-\frac{2}{v})}$ for $v\geq 3$, then we have

$(10nc_{1}c_{3})\cdot\frac{tF_{v}^{2}}{\eta_{v}}+(4c_{2}c_{3}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4c_{3}\eta_{v}^{2})\cdot\frac{tF_{v}}{\eta_{v}}\leq
tF_{v}^{\frac{v+1}{v}},\ v\geq 3.$

So we let $F_{v+1}=F_{v}^{\frac{v+1}{v}}$, and by
$\lvert\partial_{1}(t\epsilon S_{v}+t^{\alpha+1}P_{v})\rvert_{r_{v},s_{v}}\leq
tE_{v}$, $\lvert\partial_{2}(t\epsilon
S_{v}+t^{\alpha+1}P_{v})\rvert_{r_{v},s_{v}}\leq tE_{v}$, we get that
$C_{v+1}:(I,\theta)\to(\hat{I},\hat{\theta})$

$\displaystyle\lvert\hat{I}-I\rvert_{r_{v+1},s_{v+1}},\
\lvert\hat{\theta}-\theta\rvert_{r_{v+1},s_{v+1}}$ $\displaystyle\leq$
$\displaystyle\ \big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}$ $\displaystyle\leq$ $\displaystyle\ \frac{\big{(}\
\frac{2nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n+1}}+\frac{8nc_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}\rho_{v}}\
\big{)}tE_{v}^{2}+(\ 4c_{2}K_{v}^{n}e^{-K_{v}\sigma_{v}}+4\eta_{v}^{2}\
)tE_{v}}{\gamma_{v+1}\sigma_{v+1}^{\tau+n+1}\rho_{v+1}}$ $\displaystyle\
\cdot\gamma_{v+1}\sigma_{v+1}^{\tau+n+1}\rho_{v+1}$ $\displaystyle\leq$
$\displaystyle\
tF_{v}^{\frac{v+1}{v}}\cdot\gamma_{v+1}\sigma_{v+1}^{\tau+n+1}\rho_{v+1}$
$\displaystyle\leq$ $\displaystyle\
tF_{v+1}\gamma_{v+1}\sigma_{v+1}^{\tau+n+1}\rho_{v+1}$ $\displaystyle\leq$
$\displaystyle\ tE_{v+1}.$

Thus, $\lvert C_{v+1}-id\rvert_{r_{v+1},s_{v+1}}\leq tE_{v+1}$. This proofs
$(ii)$ in Lemma 3.2 (Iteration Lemma).

So, we complete the proof of the whole Iteration Lemma. $\square$

###  3.3 Proof of convergence

Note: As $F_{v+1}=F_{v}^{\frac{v+1}{v}}$ for $v\geq 3$, we have
$F_{v}=(((F_{3}^{\frac{4}{3}})^{\frac{5}{4}})^{\frac{6}{5}}\dots)^{\frac{v}{v-1}}=F_{3}^{\frac{v}{3}}$,
then
$\eta_{v}=\frac{1}{12c_{3}}F_{v}^{\frac{1}{v}}=\frac{1}{12c_{3}}F_{3}^{\frac{1}{3}}$,
which is a constant.

Let $c_{4}=\frac{1}{12c_{3}}F_{3}^{\frac{1}{3}}=\frac{1}{3}\cdot
2^{-(\tau+\bar{n}+n+4)}F_{3}^{\frac{1}{3}}$, which is a constant, then by
$r_{v}=\eta_{v-1}r_{v-1}$, we have $r_{v}=c_{4}r_{v-1}$. In combination with
the conditions mentioned earlier, we can get the final complete parameter
settings:

$\begin{array}[]{ccc}\tau_{v}=\frac{1^{-2}+\dots+v^{-2}}{2\sum\limits_{v=1}^{\infty}v^{-2}},&s_{v}=\frac{1}{4}(1-\tau_{v})s_{0},&\sigma_{v}=\frac{1}{4}(s_{v}-s_{v+1}),\\\
K_{v+1}=4K_{v},&r_{v+1}=c_{4}r_{v},&\rho_{v}=\frac{1}{4}(r_{v}-r_{v+1}),\\\
\gamma_{v}=\frac{\gamma^{\bar{n}+1}}{2^{(\bar{n}+1)v}},&F_{v}=\frac{E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n+1}\rho_{v}},&F_{v+1}=F_{v}^{\frac{v+1}{v}},\end{array}$

where $c_{4}=\frac{1}{3}\cdot 2^{-(\tau+\bar{n}+n+4)}F_{3}^{\frac{1}{3}}$.

Besides, as $\epsilon$ and $t$ are small enough, we can get that $F_{0}$(as
well as $F_{3}$) is small enough and $K_{0}$ (as well as $K_{3}$) is big
enough, then we can set appropriate initial values, so that the initial value
conditions mentioned above are satisfied.

Going back to the Lemma 3.2 (Iteration Lemma), we have

$\displaystyle\lvert\Phi_{v}-id\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert\Phi_{v+1}-\Phi_{v}\rvert\leq\sum\limits_{v=0}^{2}\lvert\Phi_{v+1}-\Phi_{v}\rvert+\sum\limits_{v=3}^{\infty}\frac{4c_{1}E_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}}$
$\displaystyle\leq
c_{1}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+\sum\limits_{v=3}^{\infty}4c_{1}F_{v}$
$\displaystyle\leq
c_{1}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+4c_{1}\sum\limits_{v=3}^{\infty}F_{3}^{\frac{v}{3}}$
$\displaystyle\leq
c_{1}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+8c_{1}F_{0}$
$\displaystyle\leq c_{2}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}),$
$\displaystyle\lvert\omega_{v}-\omega\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert\omega_{v+1}-\omega_{v}\rvert\leq\sum\limits_{v=0}^{2}\lvert\omega_{v+1}-\omega_{v}\rvert+\sum\limits_{v=3}^{\infty}2E_{v}$
$\displaystyle\leq
c_{3}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+\sum\limits_{v=3}^{\infty}2F_{v}$
$\displaystyle\leq
c_{3}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+2\sum\limits_{v=3}^{\infty}F_{3}^{\frac{v}{3}}$
$\displaystyle\leq
c_{3}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})+4F_{0}$
$\displaystyle\leq c_{4}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}),$
$\displaystyle\lvert C_{v}-id\rvert$ $\displaystyle\leq tE_{v}\leq
tF_{v}=t(F_{3}^{\frac{1}{3}})^{v}\leq
t\big{(}c_{5}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})\big{)}^{v},$

for $v=0,1,2,3,\dots$, where $c_{2}^{{}^{\prime}},c_{4}^{{}^{\prime}}$ and
$c_{5}^{{}^{\prime}}$ depend only on
$\tau,n,\bar{n},\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$. So, as $\epsilon$
and $t$ are small enough, all of them are small enough.

Let $\Phi_{\epsilon,t}=\lim\limits_{v\to\infty}\Phi_{v}$,
$\omega_{\epsilon,t}=\lim\limits_{v\to\infty}\omega_{v}$,
$R_{\epsilon,t}(p,q)=(p,q+t\omega_{\epsilon,t})$ and
$V_{\epsilon,t}=\bigcap\limits_{v}V_{\gamma,t,v}$, then
$\Phi_{\epsilon,t}^{-1}\circ
G_{H^{\epsilon}}^{t}\circ\Phi_{\epsilon,t}=R_{\epsilon,t}$, where
$R_{\epsilon,t}$ is a rotation on $V_{\epsilon,t}\times\mathbb{T}^{n}$ with
frequency $t\omega_{\epsilon,t}$, so

$\displaystyle\lvert\Phi_{\epsilon,t}-id\rvert\leq
c_{2}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}),$
$\displaystyle\lvert\omega_{\epsilon,t}-\omega\rvert\leq
c_{4}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}).$

Let the time step be $t_{1}$ and $t_{2}$ respectively, and the corresponding
symbols of the system be added $t_{1}$ and $t_{2}$ as superscripts, similarly
constructing $\psi^{t_{1}}_{v}$ and $\psi^{t_{2}}_{v}$, then for $\xi\in
V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}}$ we have

$\displaystyle\lvert\partial_{1}\psi^{t_{1}}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{t_{1}\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{1}(t_{1}\epsilon
H_{v}+t_{1}^{\alpha+1}P_{v})\rvert_{s_{v}},$
$\displaystyle\lvert\partial_{2}\psi^{t_{1}}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{t_{1}\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{2}(t_{1}\epsilon
H_{v}+t_{1}^{\alpha+1}P_{v})\rvert_{s_{v}},$
$\displaystyle\lvert\partial_{1}\psi^{t_{2}}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{t_{2}\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{1}(t_{2}\epsilon
H_{v}+t_{2}^{\alpha+1}P_{v})\rvert_{s_{v}},$
$\displaystyle\lvert\partial_{2}\psi^{t_{2}}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{t_{2}\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{2}(t_{2}\epsilon
H_{v}+t_{2}^{\alpha+1}P_{v})\rvert_{s_{v}}.$

Comparing these two systems, combined with corollary 3.2 and corollary 3.3, we
have

$\lvert\Psi_{v}^{t_{1}}-\Psi_{v}^{t_{2}}\rvert_{D_{3}}\leq\frac{2c_{1}\cdot
2nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha})}{\gamma_{v}\sigma_{v}^{\tau+n}}.$

So,

$\lvert\Phi_{v}^{t_{1}}-\Phi_{v}^{t_{2}}\rvert\leq
2c_{2}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),\
\lvert\omega_{v}^{t_{1}}-\omega_{v}^{t_{2}}\rvert\leq
2c_{4}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),$

thus

$\lvert\Phi_{\epsilon,t_{1}}-\Phi_{\epsilon,t_{2}}\rvert\leq
2c_{2}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),\
\lvert\omega_{\epsilon,t_{1}}-\omega_{\epsilon,t_{2}}\rvert\leq
2c_{4}^{{}^{\prime}}nM_{2}l_{*}(t_{1}^{\alpha}-t_{2}^{\alpha}),$

where $M_{2}$ and $l_{*}$ are constants, and $c_{2}^{{}^{\prime}}$ and
$c_{4}^{{}^{\prime}}$ are constants depending only on
$\tau,n,\bar{n},\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$.

As for $(ii)$ in theorem 2.1 , we place the measure estimations in the next
subsection.

###  3.4 Measure Estimation

Let $V_{\epsilon,t}=\bigcap\limits_{v}V_{\gamma,t,v}$, where

$V_{\gamma,t,v}=\big{\\{}\xi\in V:\lvert e^{i\langle
k,t\omega_{v}(\xi)\rangle}-1\rvert\geq\frac{t\gamma_{v}}{{\lvert
k\rvert}^{\tau}},\forall\ k\in\mathbb{Z}^{n}\setminus\\{0\\}\big{\\}}.$

We focus on

$\big{\\{}\xi\in V:\lvert\langle k,t\omega_{v}(\xi)\rangle-2\pi
l\rvert\geq\frac{t\gamma_{v}}{{\lvert k\rvert}^{\tau}},\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\},\forall\ l\in\mathbb{Z}\big{\\}}.$

It is easy to know that the latter is contained in the former. Before
estimating its measure, we first introduce the following lemma.

######  Lemma 3.4.

(Lemma 4.9 in [5] ) Let $K\subseteq\mathbb{R}^{n}$ be compact with positive
diameter $d:=\sup_{x,y\in K}|x-y|_{2}>0$, define
$\mathcal{B}:=({K+\theta})\cap\mathbb{R}^{n}\subseteq\mathbb{R}^{n}$ for some
$\theta>0$, and $g\in C^{u_{0}+1}(\mathcal{B},\mathbb{R})$ be a function with

$\min_{y\in K}\max_{0\leq v\leq{u_{0}}}\lvert D^{v}g(y)\rvert\geq\beta,$ (3.8)

for some ${u_{0}}\in\mathbb{N}$ and $\beta>0$. Then for any $\tilde{g}\in
C^{u_{0}}(\mathcal{B},\mathbb{R})$ satisfying
$\lvert\tilde{g}-g\rvert^{u_{0}}_{\mathcal{B}}:=\max\limits_{0\leq
v\leq{u_{0}}}|D^{v}(\tilde{g}-g)|_{\mathcal{B}}\leq\frac{1}{2}\beta$, we have
the estimate

$\big{\lvert}\\{y\in K:\lvert\tilde{g}(y)\rvert\leq\epsilon\\}\big{\rvert}\leq
Bd^{n-1}(n^{-\frac{1}{2}}+2d+\theta^{-1}d)(\frac{\epsilon}{\beta})^{\frac{1}{u_{0}}}\frac{1}{\beta}\max_{0<v\leq{u_{0}}}\lvert
D^{v}g\rvert_{\mathcal{B}},$ (3.9)

whenever $0<\epsilon\leq\frac{\beta}{2u_{0}+2}$. Here, $B=3(2\pi
e)^{\frac{n}{2}}(u_{0}+1)^{u_{0}+2}[(u_{0}+1)!]^{-1}$.

######  Remark 3.1.

This Lemma is just the Lemma 4.9 in [5], and it comes from Theorem 17.1 in [4]
given by Rüssmann. The proof is similar to that.

Firstly, combined with Remark 1.1, we know that $t\omega$ satisfies the
Rüssmann’s non-degeneracy condition, and by (1.7), there exist
$\bar{n}=\bar{n}(\omega,V)\in\mathbb{N}$ and $\beta=\beta(\omega,V)>0$ such
that $\min\limits_{\xi\in V}\max\limits_{0\leq v\leq\bar{n}}|D^{v}\langle
k,t\omega(\xi)\rangle|\geq t\beta\lvert k\rvert$ for $\forall\
k\in\mathbb{Z}^{n}\setminus\\{0\\}$.

As for $\lvert\langle k,t\omega\rangle-2\pi l\rvert$, let
$\widetilde{V}=V\times(0,1)$, $\widetilde{\xi}=(\xi,\xi^{{}^{\prime}})$,
$\xi^{{}^{\prime}}\in(0,1)$, $t\widetilde{\omega}=(t\omega,-2\pi)$,
$\widetilde{k}=(k,l)$, then we have
$rank\big{\\{}\partial^{i}_{\xi}\widetilde{\omega}(\widetilde{\xi}):|i|\leq\bar{n}+1\big{\\}}=n+1,\
\widetilde{\xi}\in\widetilde{V}$, and
$\min\limits_{\widetilde{\xi}\in\widetilde{V}}\max\limits_{0\leq
v\leq\bar{n}+1}|D^{v}\langle\widetilde{k},t\widetilde{\omega}(\widetilde{\xi})\rangle|\geq
t\beta\lvert\widetilde{k}\rvert$.

Let $\mathcal{B}=(\widetilde{V}+\kappa)\cap\mathbb{R}^{n+1}$, as
$\lvert\omega_{v}-\omega\rvert\leq
c_{4}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{s})$, then for $\epsilon$ and
$t$ small enough, we have
$\lvert\langle\widetilde{k},t\widetilde{\omega}_{v}\rangle-\langle\widetilde{k},t\widetilde{\omega}\rangle\rvert^{\bar{n}+1}_{\mathcal{B}}\leq\frac{1}{2}t\beta\lvert\widetilde{k}\rvert$.
And for $\gamma_{0}$ small enough, we have $t\gamma_{v}\lvert
k\rvert^{-\tau}\leq\frac{t\beta\lvert\widetilde{k}\rvert}{2(\bar{n}+1)+2}$,
then we can use Lemma 3.4, and we have

$\displaystyle\big{\lvert}\\{\xi\in V:\lvert\langle k,t\omega_{v}\rangle-2\pi
l\rvert<t\gamma_{v}\lvert
k\rvert^{-\tau}\\}\big{\rvert}=\big{\lvert}\\{\widetilde{\xi}\in\widetilde{V}:\lvert\langle\widetilde{k},t\widetilde{\omega}_{v}\rangle\rvert<t\gamma_{v}\lvert
k\rvert^{-\tau}\\}\big{\rvert}$ $\displaystyle\ \leq\
c_{5}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{t\gamma_{v}}{t\beta})^{\frac{1}{\bar{n}+1}}\lvert
k\rvert^{\frac{-\tau}{\bar{n}+1}}\lvert\widetilde{k}\rvert^{\frac{-1}{\bar{n}+1}}\frac{1}{t\beta}\max_{0<v\leq\bar{n}+2}\lvert
D^{v}t\widetilde{\omega}\rvert_{\mathcal{B}},$

for $k\in\mathbb{Z}^{n}\setminus\\{0\\},l\in\mathbb{Z}$ , where $c_{5}=3(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}$, $d$ is the
diameter of $V\times(0,1)$. Note that for such $l$, we have $\lvert
l\rvert\leq\lvert k\rvert\hat{K}$, where
$\hat{K}=\lvert\omega\rvert_{V}+c_{4}^{{}^{\prime}}(M_{1}+2nM_{2}l_{*})+1$.
Define

$\displaystyle R^{k,l}_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau}\big{\\}},\
\lvert l\rvert\leq\lvert k\rvert\hat{K},\ k\in\mathbb{Z}^{n}\setminus\\{0\\},$
$\displaystyle R^{k}_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau},\exists\
l\in\mathbb{Z}\big{\\}},\ k\in\mathbb{Z}^{n}\setminus\\{0\\},$ $\displaystyle
R_{v}$ $\displaystyle=\big{\\{}\xi\in V:\lvert\langle
k,t\omega_{v}\rangle-2\pi l\rvert<t\gamma_{v}\lvert k\rvert^{-\tau},\exists\
l\in\mathbb{Z},\exists\ k\in\mathbb{Z}^{n}\setminus\\{0\\}\big{\\}},$

i.e.,
$R_{v}=\bigcup\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}R^{k}_{v}=\bigcup\limits_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\bigcup\limits_{\lvert
l\rvert\leq\lvert k\rvert\hat{K}}R^{k,l}_{v}$. Besides,
$\lvert\widetilde{k}\rvert^{\frac{-1}{\bar{n}+1}}\leq 1$ for all
$k\in\mathbb{Z}^{n}\setminus\\{0\\}$. Then, we have

$\displaystyle\lvert R^{k,l}_{v}\rvert$ $\displaystyle\leq
c_{5}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{t\gamma_{v}}{t\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{t\beta}\max_{0<v\leq\bar{n}+2}\lvert
D^{v}t\widetilde{\omega}\rvert_{\mathcal{B}}\lvert
k\rvert^{\frac{-\tau}{\bar{n}+1}}$ $\displaystyle\leq
c_{5}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\max_{0<v\leq\bar{n}+2}\lvert
D^{v}\omega\rvert_{\mathcal{B}}\lvert k\rvert^{\frac{-\tau}{\bar{n}+1}},$
$\displaystyle\lvert R^{k}_{v}\rvert$
$\displaystyle=\lvert\bigcup\limits_{\lvert l\rvert\leq\lvert
k\rvert\hat{K}}R^{k,l}_{v}\rvert\leq 2\lvert k\rvert\hat{K}\lvert
R^{k,l}_{v}\rvert$ $\displaystyle\leq
2\hat{K}c_{5}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}\lvert
k\rvert^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$ $\displaystyle\leq
c_{6}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}\lvert
k\rvert^{\frac{\bar{n}+1-\tau}{\bar{n}+1}},$

where $c_{6}=6(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\hat{K}$.

By $V_{\epsilon,t}=\bigcap\limits_{v=0}^{\infty}V_{\gamma,t,v}$, we have the
following inequality.

$\displaystyle\lvert V\setminus
V_{\epsilon,t}\rvert\leq\sum_{v=0}^{\infty}\lvert R_{v}\rvert\
\leq\sum_{v=0}^{\infty}\sum_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\lvert
R_{v}^{k}\rvert\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}\lvert
R_{v}^{r}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}c_{6}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)(\frac{\gamma_{v}}{\beta})^{\frac{1}{\bar{n}+1}}\frac{1}{\beta}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$
$\displaystyle\leq\sum^{\infty}_{r=1}2n(2r+1)^{n-1}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}c_{6}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)$
$\displaystyle\
\cdot\sum_{v=0}^{\infty}\gamma_{v}^{\frac{1}{\bar{n}+1}}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}.$

And for $\tau\geq(n+2)(\bar{n}+1)$,
$\bar{c}:=\sum\limits_{r=1}^{\infty}2n(2r+1)^{n-1}r^{\frac{\bar{n}+1-\tau}{\bar{n}+1}}$
is convergent. Besides,
$\gamma_{v}=\frac{\gamma^{\bar{n}+1}}{2^{(\bar{n}+1)v}}$, we have
$\sum\limits_{v=0}^{\infty}\gamma_{v}^{\frac{1}{\bar{n}+1}}=\sum\limits_{v=0}^{\infty}\frac{\gamma}{2^{v}}=2\gamma$,
and $\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}<\infty$ by analysis,
therefore,

$\displaystyle\lvert V\setminus V_{\epsilon,t}\rvert$
$\displaystyle\leq\bar{c}\cdot 2\gamma\cdot
c_{6}d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d)\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$
$\displaystyle\leq c_{7}\gamma d^{n}((n+1)^{-\frac{1}{2}}+2d+\kappa^{-1}d),$

where $c_{7}=12\bar{c}(2\pi
e)^{\frac{n+1}{2}}(\bar{n}+2)^{\bar{n}+3}[(\bar{n}+2)!]^{-1}\hat{K}\beta^{-\frac{\bar{n}+2}{\bar{n}+1}}\lvert\omega\rvert^{\bar{n}+2}_{\mathcal{B}}$,
$\bar{c}$ depends only on $n$, $\bar{n}$, $\tau$. So for $\gamma$ small enough
, $V_{\epsilon,t}$ can be a set of positive measures. And for $\gamma\to 0$,
we have $\lvert V\setminus V_{\epsilon,t}\rvert\to 0$. Then $(ii)$ in theorem
2.1 proves.

Similarly, we know that $V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}}$ is also a
set of positive measure if $\gamma$ is small enough. Since $\lvert
V\setminus(V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}})\rvert\leq\lvert
V\setminus V_{\epsilon,t_{1}}\rvert+\lvert V\setminus
V_{\epsilon,t_{2}}\rvert$, we also have $\lvert
V\setminus(V_{\epsilon,t_{1}}\cap V_{\epsilon,t_{2}})\rvert\to 0$ as
$\gamma\to 0$ .

The proof of Theorem 2.1 is complete. $\square$

##  4 Kolmogorov’s non-degeneracy condition

######  Remark 4.1.

Without the Rüssmann’s non-degeneracy condition, Theorem 2.1 also holds for
$\omega$ satisfying the Kolmogorov’s non-degeneracy condition
$\Theta_{1}\lvert\xi_{1}-\xi_{2}\rvert\leq\lvert\omega(\xi_{1})-\omega(\xi_{2})\rvert\leq\Theta_{2}\lvert\xi_{1}-\xi_{2}\rvert$,
where $\Theta_{1}$ and $\Theta_{2}$ are constants. Moreover, when time step
$t$ and disturbance parameter $\epsilon$ are small enough, both the generating
function representation of symplectic algorithm (1.2) and the generating
function representation of phase flow in nearly integrable Hamiltonian system
(1.3) have invariant tori on corresponding sets of large measures, and there
is only a slight deformation in the common area. The Hausdorff’s distance
between the invariant tori in the phase spaces of the two systems is about
$o(t^{\alpha})$.

We explain this remark in three steps.

Step1:

To be specific, if $\tau>n+2$, $\gamma_{v}$ is changed to
$\frac{\gamma}{2^{v}}$, $c_{3}$ and $c_{4}$ are changed to $2^{\tau+n+2}$ and
$\frac{1}{3}\cdot 2^{-(\tau+n+4)}F_{3}^{\frac{1}{3}}$ respectively, and the
other parameters reduce the dependence on $\bar{n}$, then we can construct a
similar iterative procedure. That is to say, there is a non-empty Cantor set
$V_{\epsilon,t}\subseteq V$ and a Whitney smooth symplectic mapping
$\Phi_{\epsilon,t}$ such that $\Phi_{\epsilon,t}^{-1}\circ
G_{H^{\epsilon}}^{t}\circ\Phi_{\epsilon,t}=R_{\epsilon,t}$, and

$\displaystyle\lvert\Phi_{\epsilon,t}-id\rvert\leq
c_{2}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}),$
$\displaystyle\lvert\omega_{\epsilon,t}-\omega\rvert\leq
c_{4}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha}),$

where $c_{2}^{{}^{\prime}}$ and $c_{4}^{{}^{\prime}}$ depend only on
$\tau,n,\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$.

As for the corresponding measure estimation, let $\frac{t\gamma}{{\lvert
k\rvert}^{\tau}}<1$, then we have

$|R^{k,l}_{v}|=|\big{\\{}\xi\in V:\lvert\langle k,t\omega_{v}\rangle-2\pi
l\rvert<t\gamma_{v}\lvert
k\rvert^{-\tau}\big{\\}}|\leq\frac{2\gamma_{v}}{|k|^{\tau+1}\Theta_{1}}.$

So,

$\lvert R^{k}_{v}\rvert\leq 2\lvert k\rvert\hat{K}\lvert
R^{k,l}_{v}\rvert\leq\frac{4\gamma_{v}\hat{K}}{|k|^{\tau}\Theta_{1}},$

here $\hat{K}$ stays the same, then, we get

$\displaystyle\lvert V\setminus V_{\epsilon,t}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\lvert R_{v}\rvert\
\leq\sum_{v=0}^{\infty}\sum_{k\in\mathbb{Z}^{n}\setminus\\{0\\}}\lvert
R_{v}^{k}\rvert\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}\lvert
R_{v}^{r}\rvert$
$\displaystyle\leq\sum_{v=0}^{\infty}\sum^{\infty}_{r=1}2n(2r+1)^{n-1}\frac{4\gamma_{v}\hat{K}}{r^{\tau}\Theta_{1}}\leq\sum^{\infty}_{r=1}2n(2r+1)^{n-1}\frac{8\gamma_{0}\hat{K}}{r^{\tau}\Theta_{1}}$
$\displaystyle\leq\bar{c}\frac{8\gamma_{0}\hat{K}}{\Theta_{1}},$

where $\bar{c}:=\sum\limits_{r=1}^{\infty}2n(2r+1)^{n-1}\frac{1}{r^{\tau}}$ is
convergent. That is to say, we also get that $V_{\epsilon,t}$ can be a set of
positive measures for $\gamma$ small enough, and $\lvert V\setminus
V_{\epsilon,t}\rvert\to 0$ as $\gamma\to 0$.

And, more importantly, in the Kolmogorov’s non-degeneracy condition, the
frequency mapping is a local differential homeomorphism. For all
$w_{*}=\omega(\xi),\xi\in V\setminus V_{\epsilon,t}$, there exist $x_{v}\in
V$, such that $\omega_{v}(x_{v})=w_{*}$, $v=0,1,2,\dots$. Let $v\to\infty$,
then there exists $x_{0}\in V$, such that
$\omega_{0}(x_{0})=w_{*}=\omega_{\infty}(x_{\infty})$, and we have

$\lvert
x_{\infty}-x_{0}\rvert\leq\frac{c_{4}^{{}^{\prime}}(M_{1}\epsilon+2nM_{2}l_{*}t^{\alpha})}{\Theta_{1}}.$

Step 2:

We can also do the same for the generating function representation of phase
flow in nearly integrable Hamiltonian system (1.3). For
$\xi\in\widetilde{V}_{\epsilon,t}$, where
$\widetilde{V}_{\epsilon,t}=\bigcap\limits_{v=0}^{\infty}\widetilde{V}_{\gamma,t,v}$,
we have

$\displaystyle\lvert\widetilde{\Phi}_{v}-id\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert\widetilde{\Phi}_{v+1}-\widetilde{\Phi}_{v}\rvert\leq\sum\limits_{v=0}^{2}\lvert\widetilde{\Phi}_{v+1}-\widetilde{\Phi}_{v}\rvert+\sum\limits_{v=3}^{\infty}\frac{4c_{1}\widetilde{E}_{v}}{\gamma_{v}\sigma_{v}^{\tau+n}}$
$\displaystyle\leq\widetilde{c}_{1}\cdot
M_{1}\epsilon+8c_{1}\widetilde{F}_{0}\leq\widetilde{c}_{2}\cdot
M_{1}\epsilon,$ $\displaystyle\lvert\widetilde{\omega}_{v}-\omega\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert\widetilde{\omega}_{v+1}-\widetilde{\omega}_{v}\rvert\leq\sum\limits_{v=0}^{2}\lvert\widetilde{\omega}_{v+1}-\widetilde{\omega}_{v}\rvert+\sum\limits_{v=3}^{\infty}2\widetilde{E}_{v}$
$\displaystyle\leq\widetilde{c}_{3}\cdot
M_{1}\epsilon+4\widetilde{F}_{0}\leq\widetilde{c}_{4}\cdot M_{1}\epsilon,$

where $\widetilde{c}_{2}$ and $\widetilde{c}_{4}$ depend only on
$\tau,n,\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$. Let
$\widetilde{\Phi}_{\epsilon,t}=\lim\limits_{v\to\infty}\widetilde{\Phi}_{v}$,
$\widetilde{\omega}_{\epsilon,t}=\lim\limits_{v\to\infty}\widetilde{\omega}_{v}$
, then we have

$\lvert\widetilde{\Phi}_{\epsilon,t}-id\rvert\leq\widetilde{c}_{2}\cdot
M_{1}\epsilon,\
\lvert\widetilde{\omega}_{\epsilon,t}-\omega\rvert\leq\widetilde{c}_{4}\cdot
M_{1}\epsilon.$

Step 3:

What is more, since the frequency mapping is a local differential
homeomorphism, we can compare the generating function representation of
symplectic algorithm (1.2) and the generating function representation of phase
flow in nearly integrable Hamiltonian system (1.3).

For the time step $t$ in the common area, we have

$\psi_{v}-\widetilde{\psi}_{v}=-\sum\limits_{0<\lvert k\rvert\leq
K_{v}}\frac{\big{(}(t\epsilon
S_{v}-t\epsilon\widetilde{S}_{v})^{*}\big{)}_{k}e^{i\langle
k,\theta\rangle}+\big{(}(t^{\alpha+1}P_{v})^{*}\big{)}_{k}e^{i\langle
k,\theta\rangle}}{e^{i\langle k,t\omega_{v}\rangle}-1},$

and

$\displaystyle\lvert\partial_{1}\psi_{v}-\partial_{1}\widetilde{\psi}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{1}(\epsilon
S_{v}-\epsilon\widetilde{S}_{v}+t^{\alpha}P_{v})\rvert_{s_{v}},$
$\displaystyle\lvert\partial_{2}\psi_{v}-\partial_{1}\widetilde{\psi}_{v}\rvert_{s_{v}-\sigma_{v}}\leq\frac{2c_{1}}{\gamma_{v}\sigma_{v}^{\tau+n}}\lvert\partial_{2}(\epsilon
S_{v}-\epsilon\widetilde{S}_{v}+t^{\alpha}P_{v})\rvert_{s_{v}}.$ (4.1)

Let
$F_{v}^{*}=\frac{E_{v}^{*}}{\gamma_{v}\sigma_{v}^{\tau+n+1}\rho_{v}},F_{v+1}^{*}=(F_{v}^{*})^{\frac{v+1}{v}}$,
$E_{0}^{*}=E_{0}-\widetilde{E}_{0}$, where $E_{0}^{*}=o(t^{\alpha})$, then for
the time step $t$ in the common area, combined with (LABEL:xy15), we have

$\displaystyle\lvert(\Phi_{v}-\widetilde{\Phi}_{v})-(\Phi_{v-1}-\widetilde{\Phi}_{v-1})\rvert\leq\frac{4c_{1}E_{v-1}^{*}}{\gamma_{v-1}\sigma_{v-1}^{\tau+n}},$
$\displaystyle\lvert(\omega_{v}-\widetilde{\omega}_{v})-(\omega_{v-1}-\widetilde{\omega}_{v-1})\rvert\leq
2E_{v}^{*}.$

Then, for the common area
$\xi\in\bigcap\limits_{v=0}^{\infty}V_{\gamma,t,v}\cap\widetilde{V}_{\gamma,t,v}$,
we have

$\displaystyle\lvert\Phi_{v}-\widetilde{\Phi}_{v}\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert(\Phi_{v+1}-\widetilde{\Phi}_{v+1})-(\Phi_{v}-\widetilde{\Phi}_{v})\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{2}\lvert(\Phi_{v+1}-\widetilde{\Phi}_{v+1})-(\Phi_{v}-\widetilde{\Phi}_{v})\rvert+\sum\limits_{v=3}^{\infty}\frac{4c_{1}E_{v}^{*}}{\gamma_{v}\sigma_{v}^{\tau+n}}$
$\displaystyle\leq c_{1}^{*}\cdot t^{\alpha}+8c_{1}F_{0}^{*}\leq
c_{2}^{*}\cdot t^{\alpha},$
$\displaystyle\lvert\omega_{v}-\widetilde{\omega}_{v}\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{\infty}\lvert(\omega_{v+1}-\widetilde{\omega}_{v+1})-(\omega_{v}-\widetilde{\omega}_{v})\rvert$
$\displaystyle\leq\sum\limits_{v=0}^{2}\lvert(\omega_{v+1}-\widetilde{\omega}_{v+1})-(\omega_{v}-\widetilde{\omega}_{v})\rvert+\sum\limits_{v=3}^{\infty}2E_{v}^{*}$
$\displaystyle\leq c_{3}^{*}\cdot t^{\alpha}+4F_{0}^{*}\leq c_{4}^{*}\cdot
t^{\alpha},$

where $c_{2}^{*}$ and $c_{4}^{*}$ depend only on
$\tau,n,\rho_{0},\sigma_{0},\gamma_{0}$ and $K_{0}$. And by
$\Phi_{\epsilon,t}-\widetilde{\Phi}_{\epsilon,t}=\lim\limits_{v\to\infty}\Phi_{v}-\widetilde{\Phi}_{v}$,
$\omega_{\epsilon,t}-\widetilde{\omega}_{\epsilon,t}=\lim\limits_{v\to\infty}\omega_{v}-\widetilde{\omega}_{v}$,
we have

$\lvert\Phi_{\epsilon,t}-\widetilde{\Phi}_{\epsilon,t}\rvert\leq
c_{2}^{*}t^{\alpha},\
\lvert\omega_{\epsilon,t}-\widetilde{\omega}_{\epsilon,t}\rvert\leq
c_{4}^{*}t^{\alpha}.$

And, for all $w_{*}=\omega(\xi),\xi\in
V_{\epsilon,t}\cap\widetilde{V}_{\epsilon,t}$, there exist
$x_{0},\widetilde{x}_{0}\in V$, such that
$\omega_{0}(x_{0})=w_{*}=\widetilde{\omega}_{0}(\widetilde{x}_{0})$, and we
have

$\lvert
x_{0}-\widetilde{x}_{0}\rvert\leq\frac{4c_{4}^{*}t^{\alpha}}{\Theta_{1}},$

here $\lvert V\setminus(V_{\epsilon,t}\cap\widetilde{V}_{\epsilon,t})\rvert\to
0$ as $\gamma\to 0$.

Thus, if time step $t$ and disturbance parameter $\epsilon$ are small enough,
$\omega$ satisfies the Kolmogorov’s non-degeneracy condition, the generating
function representation of symplectic algorithm (1.2) has invariant tori on a
set of large measures. The generating function representation of phase flow in
nearly integrable Hamiltonian system (1.3) also has invariant tori on a set of
large measures, and there is only a slight deformation in the common area. The
Hausdorff’s distance between the invariant tori in the phase spaces of the two
systems is about $o(t^{\alpha})$.

######  Remark 4.2.

There are still some problems to be studied, such as the effect of
$\Theta_{1}$ in Kolmogorov’s non-degeneracy condition in Remark 4.1 and
$\beta$ in Rüssmann’s non-degeneracy condition in Remark 1.1 on the whole
system and invariant tori, which will be considered in the subsequent study.

######  Remark 4.3.

Notice that not all sufficiently small time step sizes are appropriate. The
structure of the set of time step sizes under the Kolmogorov’s non-degeneracy
condition is given by [6]. It turns out that the set of time step sizes is a
Cantor-like set, and the density of the Cantor set is 1 at the origin of the
real line. Although the frequency vector will drift under the Rüssmann’s non-
degeneracy condition, we can get very similar results after an analysis
exactly like that in [6]. That is, the set is also a Cantor set and the
density of which is also 1 at the origin of the real line. For details, please
refer to [6].

## References

  * [1] Shang, Zaijiu.  KAM theorem of symplectic algorithms for Hamiltonian systems.  Numerische Mathematik, 83(3):477–496, 1999.
  * [2] Xu, Junxiang and You, Jiangong and Qiu, Qingjiu.  Invariant tori for nearly integrable Hamiltonian systems with degeneracy.  Mathematische Zeitschrift, 226(3):375–387, 1997.
  * [3] Pöschel, Jürgen.  A lecture on the classical KAM theorem.  Proceedings of Symposia in Pure Mathematics, 69:707–732, 2001.
  * [4] Rüssmann, Helmut.  Invariant tori in non-degenerate nearly integrable Hamiltonian systems.  Regular and Chaotic Dynamics, 6(2):119–204, 2001.
  * [5] Ding, Zhaodong and Shang, Zaijiu.  Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems.  Science China Mathematics, 61(9):1567–1588, 2018.
  * [6] Shang, Zaijiu.  Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems.  Nonlinearity, 13(1):299–308, 2000.