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Periodic points and homoclinic classes

Published online by Cambridge University Press:  20 December 2006

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

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.

Type
Research Article
Copyright
2006 Cambridge University Press

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