Marie-Claude Arnaud gave a talk in Orsay (groupe de travail théorie ergodique) on Aubry-Mather sets for exact symplectic diffeomorphisms of the annulus that deviate the vertical. These are non-empty, compact invariant subsets, which are graphs of Lipschitz functions over a subset of the circle. The set of their rotation number is all of .
M.-C. Arnaud proved recently that an Aubry-Mather set with irrational rotation number is
- differentiable on a set of either zero or full measure (with respect to its unique invariant probability);
- almost everywhere differentiability is equivalent to the measure having no zero exponent;
- regularity is equivalent to uniform hyperbolicity (according to Le Calvez this happens for an open and dense set of rotation numbers under a generic condition).
She uses Green bundles that she introduced before: they are here push forwards of the vertical line in the tangent space and that their non-coincidence implies no zero exponent.