JournalsrmiVol. 32, No. 4pp. 1295–1310

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

  • Marie-Claude Arnaud

    Université d'Avignon, France
  • Pierre Berger

    Institut Galilée, Université Paris 13, Villetaneuse, France
The non-hyperbolicity of irrational invariant curves for twist maps and all that follows cover
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Abstract

The key lemma of this article is: if a Jordan curve γ\gamma is invariant by a given C1+αC^{1+\alpha}-diffeomorphism ff of a surface and if γ\gamma carries an ergodic hyperbolic probability μ\mu, then μ\mu is supported on a periodic orbit.

From this lemma we deduce three new results for the C1+αC^{1+\alpha} symplectic twist maps ff of the annulus:

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

  2. if μ\mu is an invariant probability that is supported in an invariant curve γ\gamma with an irrational rotation number, then γ\gamma is C1C^1 μ\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 (pn/qn)({p_n}/{q_n}) is a sequence of rational numbers converging to an irrational number ω\omega; let (fk(xn))1kqn(f^k(x_n))_{1 \le k \le q_n} be a minimizing periodic orbit with rotation number pn/qn{p_n}/{q_n} and let us denote by Rn\mathcal R_n its mean residue Rn=1/2Tr(Dfqn(xn))/41/qn\mathcal R_n=\left|1/2-\mathrm {Tr}(Df^{q_n}(x_n))/4\right|^{1/q_n}. Then, if lim supn+Rn>1_{n \to +\infty} \mathcal R_n>1, the Aubry–Mather set with rotation number ω\omega is not supported in an invariant curve.

Cite this article

Marie-Claude Arnaud, Pierre Berger, The non-hyperbolicity 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