On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions

Pages: 179 - 226, Volume 16, Issue 1, September 2006

doi:10.3934/dcds.2006.16.179       Abstract        Full Text (491.7K)       Related Articles

Enrique R. Pujals - IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil (email)

Abstract: In the present paper it is proved that given a maximal invariant attracting homoclinic class for a smooth three dimensional Kupka-Smale diffeomorphism, either the diffeomorphisms is $C^1$ approximated by another one exhibiting a homoclinic tangency or a heterodimensional cycle, or it follows that the homoclinic class is conjugate to a hyperbolic set (in this case we say that the homoclinic class is "topologically hyperbolic").
    We also characterize, in any dimension, the dynamics of a topologically hyperbolic homoclinic class and we describe the continuation of this homoclinic class for a perturbation of the initial system.
    Moreover, we prove that, under some topological conditions, the homoclinic class is contained in a two dimensional manifold and it is hyperbolic.

Keywords:  Dominated splittings, uniformly hyperbolic systems, bifurcations connected with nontransversal intersection.
Mathematics Subject Classification:  Primary: 37C05 , 37D05, 37G25, 37D30 ; Secondary: 37C70, 37C75.

Received: August 2005;      Revised: March 2006;      Published: June 2006.