## Abstract

We prove that nontrivial homoclinic classes of C^{r}-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 language | English (US) |
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Pages (from-to) | 699-705 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 132 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2004 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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