Robust transitivity and topological mixing for 𝐶¹-flows

Abdenur, Flavio; Avila, Artur; Bochi, Jairo

doi:10.1090/S0002-9939-03-07187-9

Robust transitivity and topological mixing for $C^1$-flows
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by Flavio Abdenur, Artur Avila and Jairo Bochi

Proc. Amer. Math. Soc. 132 (2004), 699-705

DOI: https://doi.org/10.1090/S0002-9939-03-07187-9

Published electronically: October 21, 2003

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Abstract:

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.

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