Ergodic Theory and Dynamical Systems



Periodic points and homoclinic classes


F. ABDENUR a1, CH. BONATTI a2, S. CROVISIER a3, L. J. DÍAZ a4 and L. WEN a5
a1 Departamento Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro, RJ, Brazil (e-mail: flavio@mat.puc-rio.br)
a2 Institut de Mathématiques de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France (e-mail: bonatti@u-bourgogne.fr)
a3 CNRS–LAGA, UMR 7539, Université Paris 13, Avenue J.-B. Clément, 93430 Villetaneuse, France (e-mail: crovisie@math.univ-paris13.fr)
a4 Departamento Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro, RJ, Brazil (e-mail: lorenzojdiaz@gmail.com)
a5 School of Mathematics, Peking University, Beijing 100871, People's Republic of China (e-mail: lwen@math.pku.edu.cn)

Article author query
abdenur f   [Google Scholar] 
bonatti ch   [Google Scholar] 
crovisier s   [Google Scholar] 
diaz lj   [Google Scholar] 
wen l   [Google Scholar] 
 

Abstract

We prove that there is a residual subset $\mathcal{I}$ of ${\rm Diff}^1({\it M})$ such that any homoclinic class of a diffeomorphism $f\in \mathcal{I}$ having saddles of indices $\alpha$ and $\beta$ contains a dense subset of saddles of index $\tau$ for every $\tau\in [\alpha,\beta]\cap \mathbb{N}$. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of $C^1$-generic diffeomorphisms.

(Published Online December 20 2006)
(Received September 1 2005)
(Revised May 16 2006)