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1 results

  • Denjoy subsystems and horseshoes

  • Edited by Albert Fathi, Georgia Institute of Technology, Philip J. Morrison, University of Texas, Austin, Tere M-Seara, Universitat Politècnica de Catalunya, Barcelona, Sergei Tabachnikov, Pennsylvania State University
  • Book:
    Hamiltonian Systems
    Published online:
    10 May 2024
    Print publication:
    09 May 2024, pp 1-28

  • Summary

    We introduce a notion of weak Denjoy subsystem (WDS) that generalizes the Aubry–Mather–Cantor sets to diffeomorphisms of manifolds. We explain how a rotation number can be associated to such a WDS. Then we build in any horseshoe a continuous one parameter family of such WDS that is indexed by its rotation number. Looking at the inverse problem in the setting of Aubry– Mather theory, we also prove that for a generic conservative twist map of the annulus, the majority of the Aubry–Mather sets are contained in some horseshoe that is associated to an Aubry–Mather set with a rational rotation number.