Robust transitivity and topological mixing for $C^1$-flows

Details Robust transitivity and topological mixing for $C^1$-flows Robust transitivity and topological mixing for $C^1$-flows

2002-02-20

arxiv.org/abs/math/0202207v3

Proceedings of American Mathematical Society, 132 (2004), 699-705.

Mathematics

Dynamical Systems

Final version, to appear in the Proceedings of the AMS

Scientific paper

10.1090/S0002-9939-03-07187-9

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.

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