Homogenization on arbitrary manifolds

We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a new simple variational proof of standard homogenization results.

Generic hyperbolicity of Aubry sets on surfaces

Given a Tonelli Hamiltonian of class 𝐶2 on the cotangent bundle of a compact surface, we show that there is an open dense set of potentials in the 𝐶2 topology for which the Aubry set is hyperbolic in its energy level.