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 Lambda, 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 Lambda(Y) of Lambda 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.