Invariant Curves of Analytic Reversible Mappings Under Brjuno–Rüssmann’s Non-resonant Condition

Abstract

In this paper we prove the existence of invariant curves for analytic reversible mappings under Brjuno–Rüssmann’s non-resonant condition. In the proof we use the polynomial structure of function to truncate, introduce a parameter $q$ and make the steps of KAM iteration infinitely small in the speed of function $q^{n}\epsilon ,0 <q<1,$ rather than super exponential function.

Appendix

In this section we formulate some lemmas which have been used in the previous section. For the detailed proofs, refer to [13, 27].

In the construction of the transformation in Lemma 2.1, we will meet the following difference equation:

$$\begin{aligned} l(x+\omega )-l(x)=g(x). \end{aligned}$$

(4.1)

Lemma 4.1

Suppose that $l(x)$ and $g(x)$ are real analytic on $D(s),$ where $D(s)=\{x\in \mathbb {C}/ 2\pi \mathbb {Z}~|~ |\text{ Im }x|\le s\}.$ Suppose $\omega$ satisfies the Brjuno–Rüssmann’s non-resonant condition $|\frac{k\omega }{2\pi }-l|\ge \frac{\alpha }{\Delta {(|k|)}}, ~~ \forall (k,l)\in \mathbb {Z}\times \mathbb {Z}{\setminus }\{0,0\},$ then the difference equation (4.1) has a unique solution $l(x)\in D(s)$ satisfying

$$\begin{aligned} \Vert l(x)\Vert _{s}\le \frac{\Delta (|k|)}{\alpha }\Vert g(x)\Vert _{s}. \end{aligned}$$

Moreover, if $g(-x-\omega )=g(x),$ then $l(x)$ is odd in $x;$ if $g(-x-\omega )=-g(x),$ $l(x)$ is even in $x.$

Lemma 4.2

Suppose $g(x)$ is m-th differentiable function on the closure $\bar{I}$ of $I,$ where $I\subset \mathbb {R}$ is an interval. Let $I_{h}=\{x~|~|g(x)|<h,x\in I\}, h>0.$ If $|g^{(m)}(x)|\ge d>0$ for all $x\in I,$ where $d$ is a constant, then

$$\begin{aligned} \text{ meas }(I_{h})\le ch^{\frac{1}{m}}, \end{aligned}$$

where $c=2(2+3+\cdots +m+d^{-1}).$