The non-hyperbolicity of irrational invariant curves for twist maps and all that follows

Abstract

The key lemma of this article is: if a Jordan curve $\gamma$ is invariant by a given $C^{1+\alpha }$-diffeomorphism $f$ of a surface and if $\gamma$ carries an ergodic hyperbolic probability $\mu$, then $\mu$ is supported on a periodic orbit.

From this lemma we deduce three new results for the $C^{1+\alpha }$ symplectic twist maps $f$ of the annulus:

1. if $\gamma$ is a loop at the boundary of an instability zone such that $f_{|\gamma }$ has an irrational rotation number, then the convergence of any orbit to $\gamma$ is slower than exponential;

2. if $\mu$ is an invariant probability that is supported in an invariant curve $\gamma$ with an irrational rotation number, then $\gamma$ is $C^1$ $\mu$-almost everywhere;

3. we prove a part of the so-called "Greene criterion", introduced by J.M. Greene in 1978 and never proved: assume that $(p_n/q_n)$ is a sequence of rational numbers converging to an irrational number $\omega$; let $(f^k(x_n){)}_{1\le k\le q_n}$ be a minimizing periodic orbit with rotation number $p_n/q_n$ and let us denote by ${\mathcal{R}}_n$ its mean residue ${\mathcal{R}}_n={\left|1/2-\mathrm{Tr}(D{f}^{q_n}(x_n))/4\right|}^{1/q_n}$. Then, if lim sup${}_{n\to +\infty }{\mathcal{R}}_n>1$, the Aubry–Mather set with rotation number $\omega$ is not supported in an invariant curve.