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


Authors: Flavio Abdenur, Artur Avila and Jairo Bochi
Journal: Proc. Amer. Math. Soc. 132 (2004), 699-705
MSC (2000): Primary 37C20
Published electronically: October 21, 2003
MathSciNet review: 2019946
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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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Additional Information

Flavio Abdenur
Affiliation: IMPA, Estr. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: flavio@impa.br

Artur Avila
Affiliation: Collège de France, 3 rue d’Ulm, 75005 Paris, France
Email: avila@impa.br

Jairo Bochi
Affiliation: IMPA, Estr. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil
Email: bochi@impa.br

DOI: https://doi.org/10.1090/S0002-9939-03-07187-9
Keywords: Generic properties of flows, homoclinic classes, topological mixing
Received by editor(s): February 8, 2002
Published electronically: October 21, 2003
Additional Notes: The first author was supported by FAPERJ and Prodoc/CAPES, the second author was supported by FAPERJ and CNPq, and the third author was supported by Profix/CNPq
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society