Posts Tagged ‘smoothness and hyperbolicity’

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 \mathbb R.

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;
  • C^1 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.

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