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

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.