We consider the irrational Aubry-Mather sets of an exact symplectic monotone C1 twist map of the two-dimensional annulus, 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 if and only if its support is C1-regular almost everywhere;
∙ a Mather measure has nonzero Lyapunov exponents if and only if its support is C1-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 C1-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.