Robust Transitivity and Topological Mixing for C 1 -flows

We prove that non-trivial homoclinic classes of C r-generic flows are topo-logically mixing. This implies that given Λ a non-trivial 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 non-trivial homoclinic classes are topologically mixing is not open and dense, in general. Throughout this paper M denotes a compact d-dimensional boundaryless manifold, d ≥ 3, and X r (M) is the space of C r vector fields on M endowed with the usual C r topology, where r ≥ 1. We shall prove that, generically (residually) in X r (M), nontrivial homoclinic classes are topologically mixing. As a consequence, nontrivial C 1-robustly transitive sets (and C 1-robustly transitive flows in particular) become topologically mixing after arbitrarily small C 1-perturbations of the flow. These results generalize the following theorem by Bowen [B]: non-trivial basic sets of C r-generic Axiom A flows are topologically mixing. Note that C 1-robustly transitive sets are a natural generalization of hyperbolic basic sets; they are the subject of several recent papers, such as [BD1] and [BDP]. In order to announce precisely our results, let us introduce some notations and definitions. Given t ∈ R and X ∈ X r (M), we shall denote by X t the induced time t map. A subset R of X r (M) is residual if it contains the intersection of a countable number of open dense subsets of X r (M). Residual subsets of X r (M) are dense. Given an open subset U of X r (M), then property (P) is generic in U if it holds for all flows in a residual subset R of U ; (P) is generic if it is generic in all of X r (M). A compact invariant set for X is non-trivial if it is neither a periodic orbit nor a single point. A compact invariant set Λ of X is transitive if it there is some point x ∈ Λ such that the future orbit {X t (x) : t > 0} of x is dense in Λ; Λ is topologically mixing …