MPEJ Volume 12, No. 3, 29 pp.
Received: Nov 2, 2005. Revised: May 8, 2005. Accepted:  Jul 15, 2006.

D. Gomes, C. Valls
Perturbation theory and discrete Hamiltonian dynamics

ABSTRACT: 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.

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