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

The key lemma of this article is: if a Jordan curve γ is invariant by a given C1+α-diffeomorphism f of a surface and if γ carries an ergodic hyperbolic probability μ, then μ is supported on a periodic orbit. From this lemma we deduce three new results for the C1+α symplectic twist maps ff of the annulus: 1) if γ is a loop at the boundary of an instability zone such that f|γ has an irrational rotation number, then the convergence of any orbit to γ is slower than exponential; 2) if μ is an invariant probability that is supported in an invariant curve γ with an irrational rotation number, then γ is C1 μμ-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) is a sequence of rational numbers converging to an irrational number ω; let (fk(xn))1≤k≤qn be a minimizing periodic orbit with rotation number pn/qn and let us denote by Rn its mean residue Rn=|1/2−Tr(Dfqn(xn))/4|1/qn. Then, if lim supn→+∞Rn>1, the Aubry–Mather set with rotation number ω is not supported in an invariant curve.