Robust transitivity and topological mixing for C¹-flows

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.