Robust transitivity and topological mixing for C1-flows

Flavio Abdenur, Artur Avila, Jairo Bochi

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Abstract

We prove that nontrivial homoclinic classes of Cr-generic flows are topologically mixing. This implies that given Λ, a nontrivial C 1-robustly transitive set of a vector field X, there is a C 1 -perturbation Y of X such that the continuation Λ Y of Λ is a topologically mixing set for Y. In particular, robustly transitive flows become topologically mixing after C 1-perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose nontrivial homoclinic classes are topologically mixing is not open and dense, in general.

Original languageEnglish (US)
Pages (from-to)699-705
Number of pages7
JournalProceedings of the American Mathematical Society
Volume132
Issue number3
DOIs
StatePublished - Mar 2004

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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