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

Abstract

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

• $\bullet$  a Mather measure has zero Lyapunov exponents if and only if its support is $C^1$-regular almost everywhere;
• $\bullet$  a Mather measure has nonzero Lyapunov exponents if and only if its support is $C^1$-irregular almost everywhere;
• $\bullet$  an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere;
• $\bullet$  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.