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A heteroclinic bifurcation of Anosov diffeomorphisms

Published online by Cambridge University Press:  01 June 1998

HEBER ENRICH
HEBER ENRICH
Affiliation:
IMERL Facultad de Ingeniería, C.C. 30 Montevideo, Uruguay (e-mail: enrich@fing.edu.uy)
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Abstract

We study some diffeomorphisms in the boundary of the set of Anosov diffeomorphisms mainly from the ergodic viewpoint. We prove that these diffeomorphisms, obtained by isotopy from an Anosov $f:M \mapsto M$ through a heteroclinic tangency, determine a manifold ${\cal M}$ of finite codimension in the set of $C^r$ diffeomorphisms. We prove that any diffeomorphism $F$ in ${\cal M}$ is conjugate to $f$; moreover, there exists a unique SRB measure for $F$, and $F$ is Bernoulli with respect to this measure. In particular, if the dimension of $M$ is two, and $\mu $ is a volume element, we prove that the isotopy can be taken such that the measure is preserved.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 3 , June 1998 , pp. 567 - 608
Copyright
© 1998 Cambridge University Press

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Footnotes

To the memory of Professor R. Mañé, who has been my advisor during the preparation of this article, a part of which has been presented as a doctoral thesis.