Persistence of Lower Dimensional Hyperbolic Invariant Tori for Nearly Integrable Symplectic Mappings

Abstract

In this paper we consider a class of nearly integrable symplectic mappings with generating function and prove the persistence of lower dimensional hyperbolic invariant tori. Under a Rüssmann-type non-degenerate condition, by using the KAM theory, we proved that the nearly integrable twist symplectic mappings admit a family of lower dimensional invariant tori as long as the generating function is real analytic and the perturbation is sufficiently small.

Appendix

This part we will give several lemmas that describes properties of the norm $\Vert \cdot \Vert _{s,r}$. The proofs are very similar to [10] and even simpler. Let $\mathcal {A}_{s,r}$ denote the space of real analytic function defined on $D(s,r)$. If $f\in \mathcal {A}_{s,r}$, then we can write it as Fourier series $f=\sum \nolimits _{k\in \mathbb {Z}^n}f_ke^{i\langle k,x\rangle }$. Let $\mathcal {A}^0_{s,r}=\{f|f\in \mathcal {A}_{x,r},[f]=0\}$, where $[f]$ denotes the average of $f$.

Lemma 2.6

(Cauchy’s estimate) Let $f\in \mathcal {A}_{s,r}$. Then $\forall \rho \in (0,s),\forall \sigma \in (0,r)$, we have

$$\begin{aligned} \begin{array}{cc} \Vert \partial _xf\Vert _{s,r}\leqslant \frac{1}{e\rho }\Vert f\Vert _{s,r}, &{} \Vert \partial _yf\Vert _{s,r-\sigma }\leqslant \frac{1}{\sigma ^2}\Vert f\Vert _{s,r}, \\ \Vert \partial _uf\Vert _{s,r-\sigma }\leqslant \frac{1}{\sigma }\Vert f\Vert _{s,r}, &{} \Vert \partial _vf\Vert _{s,r-\sigma }\leqslant \frac{1}{\sigma }\Vert f\Vert _{s,r}. \end{array} \end{aligned}$$

Lemma 2.7

Let $f\in \mathcal {A}_{s,r}.$ We write it as Fourier series$~~f=\sum _{k\in \mathbb {Z}^n}f_ke^{i\langle k,x\rangle }$. Then we have

$$\begin{aligned} |f_k|\leqslant \Vert f\Vert _{s,r}e^{-|k|s},\quad \forall k\in \mathbb {Z}^n, \end{aligned}$$

where$|k|=|k_1|+|k_2|+\cdots +|k_n|.$

Below we give a theorem of invertible mappings.

Lemma 2.8

Let $f\in \mathcal {A}_{s,r}$,$T_K(f)=\sum \nolimits _{|k|<K}f_ke^{i\langle k,x\rangle }$. Then we have

$$\begin{aligned} \Vert f-T_K(f)\Vert _{s-\rho ,r}\le c K^ne^{-K\rho }\Vert f\Vert _{s,r},~~~0<\rho <s, \end{aligned}$$

where the constant c only depends on n.

Lemma 2.9

Let $\Omega _h$ is complex neighborhood of $\Omega$ with the radius $h>0.$ Suppose $f:\Omega _h \rightarrow C^n$ is an analytic mapping and

$$\begin{aligned} |f-id |_{\Omega _h}\le \epsilon <h/2. \end{aligned}$$

where $|\cdot |_{\Omega _h}$ indicates the super-norm on $\Omega _h.$

Then $f$ has an analytic inverse $\phi : \Omega _{h-2\epsilon } \rightarrow \Omega _h$ such that $|\phi -id |_{\Omega _{h-2\epsilon }}\le \epsilon .$