Journal of the Institute of Mathematics of Jussieu



ROBUST HETERODIMENSIONAL CYCLES AND $C^1$-GENERIC DYNAMICS


Christian Bonatti a1 and Lorenzo J. Díaz a2
a1 Institut de Mathématiques de Bourgogne, BP 47 870, 21078 Dijon Cedex, France (bonatti@u-bourgogne.fr)
a2 Departamento Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro, Brazil (lodiaz@mat.puc-rio.br)

Article author query
bonatti c   [Google Scholar] 
díaz lj   [Google Scholar] 
 

Abstract

A diffeomorphism $f$ has a heterodimensional cycle if there are (transitive) hyperbolic sets $\varLambda$ and $\varSigma$ having different indices (dimension of the unstable bundle) such that the unstable manifold of $\varLambda$ meets the stable one of $\varSigma$ and vice versa. This cycle has co-index $1$ if $\mathop{\mathrm{index}}(\varLambda)=\mathop{\mathrm{index}}(\varSigma)\pm1$. This cycle is robust if, for every $g$ close to $f$, the continuations of $\varLambda$ and $\varSigma$ for $g$ have a heterodimensional cycle.

We prove that any co-index $1$ heterodimensional cycle associated with a pair of hyperbolic saddles generates $C^1$-robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.

We also derive some consequences from this result for $C^1$-generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle.

(Published Online December 10 2007)
(Received December 19 2006)
(Accepted June 18 2007)


Key Words: axiom A; chain recurrence class; dominated splitting; heterodimensional cycle; homoclinic class; hyperbolicity.

Maths Classification

Primary 37C20; 37C29; 37C70; Secondary 37C25.


Dedication:
Dedicated to Welington de Melo for his 60th birthday