The key lemma of this article is: if a Jordan curve is invariant by a given -diffeomorphism of a surface and if carries an ergodic hyperbolic probability , then is supported on a periodic orbit.
From this lemma we deduce three new results for the symplectic twist maps of the annulus:
if is a loop at the boundary of an instability zone such that has an irrational rotation number, then the convergence of any orbit to is slower than exponential;
if is an invariant probability that is supported in an invariant curve with an irrational rotation number, then is -almost everywhere;
we prove a part of the so-called "Greene criterion", introduced by J.M. Greene in 1978 and never proved: assume that is a sequence of rational numbers converging to an irrational number ; let be a minimizing periodic orbit with rotation number and let us denote by its mean residue . Then, if lim sup, the Aubry–Mather set with rotation number 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. 1295–1310DOI 10.4171/RMI/917