Heterodimensional cycles, partial hyperbolicity and limit dynamics

L. J. Diaz ¹   ; J. Rocha ²

¹ Depto. Matemática PUC-Rio Marquês de S. Vicente núm. 225 22453-900 Rio de Janeiro RJ, Brazil
² Departamento de Matemática Pura Universidade do Porto Rua do Campo Alegre núm. 687 4169-007 Porto, Portugal

Fundamenta Mathematicae, Tome 174 (2002) no. 2, pp. 127-186

Résumé

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.

Détail

DOI : 10.4064/fm174-2-2

Keywords: study one parameter families diffeomorphisms unfolding heterodimensional cycles cycles containing periodic points different indices construct set arcs subset parameter space positive relative density bifurcation value resulting nonwandering set disjoint union hyperbolic basic sets different indices strong partially hyperbolic set which robustly transitive dynamics diffeomorphisms consider partially hyperbolic one dimensional central direction main tool proving results construction one dimensional model given iterated function system which describes limit dynamics central direction selected parameters arc translate properties model family diffeomorphisms