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

We consider the irrational Aubry-Mather sets of an exact symplectic monotone C 1 twist map, introduce for them a notion of " C 1-regularity " (related to the notion of Bouligand paratingent cone) and prove that : • a Mather measure has zero Lyapunov exponents iff its support is almost everywhere C 1-regular; • a Mather measure has non zero Lyapunov exponents iff its support is almost everywhere C 1-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 C 1-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.