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On the existence of non-trivial homoclinic classes

Published online by Cambridge University Press:  01 October 2007

CHRISTIAN BONATTI ,
SHAOBO GAN  and
LAN WEN
CHRISTIAN BONATTI
Affiliation:
IMB, UMR 5584 du CNRS, BP 47870, 21078 Dijon Cedex, France (email: bonatti@u-bourgogne.fr)
SHAOBO GAN
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China (email: gansb@math.pku.edu.cn, lwen@math.pku.edu.cn)
LAN WEN
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China (email: gansb@math.pku.edu.cn, lwen@math.pku.edu.cn)
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Abstract

We show that, for C1-generic diffeomorphisms, every chain recurrent class C that has a partially hyperbolic splitting $E^s\oplus E^c\oplus E^u$ with dimEc=1 either is an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also prove that, for C1-generic diffeomorphisms, any chain recurrent class that has a dominated splitting $E\oplus F$ with dim(E)=1 either is a homoclinic class, or the bundle E is uniformly contracting. As a corollary we prove in dimension three a conjecture of Palis, which announces that any C1-generic diffeomorphism is either Morse–Smale, or has a non-trivial homoclinic class.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1473 - 1508
Copyright
Copyright © Cambridge University Press 2007

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