Preprint A418/2005
On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphism in attracting regions

Enrique R. Pujals

**Keywords: **

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.