Revista Matemática Iberoamericana

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Volume 32, Issue 4, 2016, pp. 1295–1310
DOI: 10.4171/RMI/917

Published online: 2016-12-16

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

Marie-Claude Arnaud[1] and Pierre Berger[2]

(1) Université d'Avignon, France
(2) Institut Galilée, Université Paris 13, Villetaneuse, France

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 $\mathcal R_n=\left|1/2-\mathrm {Tr}(Df^{q_n}(x_n))/4\right|^{1/q_n}$. 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.

Keywords: Symplectic dynamics, twist maps, Aubry–Mather theory, Green bundles, Greene criterion, Lyapunov exponents, invariant curves, instability zones

Arnaud Marie-Claude, Berger Pierre: The non-hyperbolicity of irrational invariant curves for twist maps and all that follows. Rev. Mat. Iberoam. 32 (2016), 1295-1310. doi: 10.4171/RMI/917