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

Flavio Abdenur·Artur Avila·Jairo Bochi
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Abstract

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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Flavio Abdenur, Artur Avila, Jairo Bochi. 2002-02-20. Robust transitivity and topological mixing for $C^1$-flows. https://doi.org/10.1090/s0002-9939-03-07187-9

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