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

Abstract

The key result of this article is key lemma: if a Jordan curve γ is invariant by a given C 1+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 C 1+α symplectic twist maps f 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 C 1 µ-almost everywhere; 3. we prove a part of the so-called criterion, introduced by J. M. Greene in [16] in 1978 and never proved: assume that (pn qn) is a sequence of rational numbers converging to an irrational number ω; let (f k (x n)) 1≤k≤qn be a minimizing periodic orbit with rotation number pn qn and let us denote by R n its mean residue R n = |1/2 − Tr(Df qn (x n))/4|