Robust transitivity and topological mixing for 𝐶¹-flows

Abstract

We prove that nontrivial homoclinic classes of C r C^r -generic flows are topologically mixing. This implies that given Λ \Lambda , a nontrivial C 1 C^1 -robustly transitive set of a vector field X X , there is a C 1 C^1 -perturbation Y Y of X X such that the continuation Λ Y \Lambda _Y of Λ \Lambda is a topologically mixing set for Y Y . In particular, robustly transitive flows become topologically mixing after C 1 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.