Persistence of Invariant Tori in Infinite-Dimensional Hamiltonian Systems

Abstract

In this paper, we consider the persistence of invariant tori in infinite-dimensional Hamiltonian systems

$$\begin{aligned} H=\langle \omega ,I \rangle +P(\theta ,I,\omega ), \end{aligned}$$

where $\theta \in \mathbb {T}^\Lambda$, $I\in \mathbb {R}^\Lambda$, the frequency ${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$ is regarded as parameters varying freely over some subset $\ell ^\infty (\Lambda ,\mathbb {R})$ of the parameter space $\mathbb {R}^\Lambda$, ${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$ is a bilateral infinite sequence of rationally independent frequency, in other words, any finite segments of ${\omega }=(\cdots ,{\omega }_\lambda ,\cdots )_{\lambda \in \Lambda }$ are rationally independent.

Appendix A: Some Useful Lemmas

In this appendix, we give some useful lemmas, which are used in proving Theorem 3.1.

Lemma A.1

(Lemma B.1 in [13])

• (i) Let $\mu _1, \mu _2>0$. Then

  $$\begin{aligned} \sup \limits _{\begin{array}{c} \ell \in \mathbb {Z}_*^\infty \\ |\ell |_\eta <\infty \end{array}}\prod \limits _{i} (1+\langle i\rangle ^{\mu _1} |\ell _i|^{\mu _2})e^{-\rho |\ell |_\eta }\le \exp \Bigg ({\tau \over {\rho ^{1\over \eta }}}\ln {\tau \over \rho }\Bigg ) \end{aligned}$$

  for some constant $\tau =\tau (\eta , \mu _1, \mu _2)>0$.

• (ii) Let $\rho >0$. Then $\sum \nolimits _{\ell \in \mathbb {Z}_*^\infty }e^{-\rho |\ell |_\eta }\lesssim \exp \Bigg ({\tau \over {\rho ^{1\over \eta }}}\ln {\tau \over \rho }\Bigg )$ for some constant $\tau =\tau (\eta )>0$.

Lemma A.2

(Cauchy estimates, Lemma 2.7 in [13]) Let $\sigma ,\rho >0$ and $u\in \mathcal {H}(\mathbb {T}_{\sigma +\rho }^\infty ,X).$ Then for any $k\in \mathbb {N}$, the kth differential $d_\varphi ^k u$ satisfies the estimate

$$\begin{aligned} \Vert d_\varphi ^k u\Vert _{\mathcal {H}(\mathbb {T}_{\sigma }^\infty ,\mathcal {M}_k)}\lesssim _k \rho ^{-k}\Vert u\Vert _{\sigma +\rho }. \end{aligned}$$

Lemma A.3

(Lemma 10 in [15]) Suppose that for some $v\ge w$,

$$\begin{aligned} \rho _0^{-1}|||F|||_{v,r_0-\rho _0,s_0},\ \ \ \sum \limits _\lambda |||F_{\theta _\lambda }|||_{v,r_0-\rho _0,s_0}\le M<{s\over 8}. \end{aligned}$$

Then

$$\begin{aligned} |||G\circ \Phi |||_{v,r-\rho ,s/2}\le {1\over {1-8M/s}}|||G|||_{v,r,s} \end{aligned}$$

for $0<\rho _0\le \rho <r\le \rho _0-s_0$ and $0<s\le s_0/2$, where $\Phi$ denotes the time-1-map of the hamiltonian vectorfield $X_F$.

Lemma A.4

(Lemma 11 in [15]) Suppose f is real analytic from ${\mathcal {W}}_h$ into $\mathbb {C}^\Lambda$. If

$$\begin{aligned} |f-id|_\infty \le \delta \le h/4 \end{aligned}$$

on ${\mathcal {W}}_h$, then f has a real analytic inverse $\varphi$ on ${\mathcal {W}}_{h/4}$. Moreover,

$$\begin{aligned} |\varphi -id|_\infty ,\ \ \ \ {h\over 4}|\partial \varphi -I|_\infty \le \delta \end{aligned}$$

on this domain.