Hostname: page-component-7bb8b95d7b-wpx69 Total loading time: 0 Render date: 2024-09-06T16:15:05.153Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.