Perturbation theory and discrete Hamiltonian dynamics

In this paper we discuss a weak version of KAM theory for symplectic maps which arise from the discretization of the minimal action principle. These maps have certain invariant sets, the Mather sets, which are the generalization of KAM tori in the non-differentiable case. These sets support invariant measures, the Mather measures, which are action minimizing measures. We generalize viscosity solution methods to study discrete systems. In particular, we show that, under non-resonance conditions, the Mather sets can be approximated uniformly, up to any arbitrary order, by finite perturbative expansions. We also present new results concerning the approximation of Mather measures.