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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics