The link between the shape of the Aubry Mather sets and their Lyapunov exponents

The link between the shape of the Aubry-Mather sets and their Lyapunov exponents M.-C. ARNAUD ?† September 2, 2008 Abstract We consider the irrational Aubry-Mather sets of an exact symplectic monotone C1 twist map, introduce for them a notion of “C1-regularity” (related to the notion of Bouligand paratingent cone) and prove that : • a Mather measure has zero Lyapunov exponents iff its support is almost every- where C1-regular; • a Mather measure has non zero Lyapunov exponents iff its support is almost everywhere C1-irregular; • an Aubry-Mather set is uniformly hyperbolic iff it is everywhere non regular; • the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be non C1-regular, are not “too irregular” (i.e. have small paratingent cones). The main tools that we use in the proofs are the so-called Green bundles. ?ANR KAM faible †Universite d'Avignon et des Pays de Vaucluse, Laboratoire d'Analyse non lineaire et Geometrie (EA 2151), F-84 018Avignon, France. e-mail: 1

• c1 regularity


• mather measure


• twist map


• avignon


• †universite d'avignon et des pays de vaucluse


• lipschitz graph


• symplectic twist


• every generic exact


• almost everywhere


• minimal invariant compact