Perturbation theory and discrete Hamiltonian dynamics

Diogo A. Gomes*, Claudia Valls

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalMathematical Physics Electronic Journal
StatePublished - 2006
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability


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