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

TL;DR

This paper proves that irrational invariant curves in twist maps are non-hyperbolic, showing that convergence to such curves is slower than exponential and confirming a part of Greene's criterion for stability.

Contribution

It introduces a key lemma linking ergodic hyperbolic measures to periodic orbits on invariant curves, and derives new results on the dynamics near irrational invariant curves in twist maps.

Findings

Convergence to boundary invariant curves with irrational rotation number is slower than exponential.

Invariant curves with irrational rotation number are C1 almost everywhere under the invariant measure.

Partial proof of Greene's criterion relating rational approximations to stability of invariant curves.

Abstract

The key result of this article is key lemma: 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 [16] in 1978 and never proved: assume that (pn qn) is a sequence of rational numbers converging to an irrational number $\omega$; let (f k (x n)) 1$\le$k$\le$qn be a minimizing periodic orbit with rotation number pn qn and let us denote by R n its mean residue R n = |1/2 -- Tr(Df qn (x n))/4|