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We consider the irrational Aubry-Mather sets of an exact symplectic monotone C 1 twist map of the two-dimensional annulus, 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 if and only if its support is C 1 -regular almost everywhere; a Mather measure has nonzero Lyapunov exponents if and only if its support is C 1 -irregular almost everywhere; an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere; the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be C 1 -irregular, 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.

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