The nonhyperbolicity of irrational invariant curves for twist maps and all that follows
MarieClaude Arnaud
Université d'Avignon, FrancePierre Berger
Institut Galilée, Université Paris 13, Villetaneuse, France
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:

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;

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;

we prove a part of the socalled "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 $\mathcal R_n=\left1/2\mathrm {Tr}(Df^{q_n}(x_n))/4\right^{1/q_n}$. 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.
Cite this article
MarieClaude Arnaud, Pierre Berger, The nonhyperbolicity of irrational invariant curves for twist maps and all that follows. Rev. Mat. Iberoam. 32 (2016), no. 4, pp. 1295–1310
DOI 10.4171/RMI/917