# American Institute of Mathematical Sciences

March  2006, 16(1): 179-226. doi: 10.3934/dcds.2006.16.179

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

 1 IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil

Received  August 2005 Revised  March 2006 Published  June 2006

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.
Citation: Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179
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