Heterodimensional cycles, partial hyperbolicity and limit dynamics

Tom 174 / 2002

L. J. Diaz, J. Rocha Fundamenta Mathematicae 174 (2002), 127-186 MSC: 37C20, 37D30. DOI: 10.4064/fm174-2-2


We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms we consider is partially hyperbolic with one-dimensional central direction. The main tool for proving our results is the construction of a one-dimensional model given by an iterated function system which describes the limit dynamics in the central direction. For selected parameters of the arc, we translate properties of the model family to the diffeomorphisms.


  • L. J. DiazDepto. Matemática PUC-Rio
    Marquês de S. Vicente núm. 225
    22453-900 Rio de Janeiro RJ, Brazil
  • J. RochaDepartamento de Matemática Pura
    Universidade do Porto
    Rua do Campo Alegre núm. 687
    4169-007 Porto, Portugal

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