Robust transitivity and topological mixing for C^1 flows

Jairo Bochi | Abdenur, Flavio | Avila, Artur

Keywords: Generic properties | homoclinic classes | topological mixing

We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $Lambda$ 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 $\Lambda_Y$ 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 non-trivial homoclinic classes are topologically mixing is \emph{not} open and dense, in general.

Anexos:

• mixing22.pdf