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# 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.
Mathematics Subject Classification: Primary: 37C05 , 37D05, 37G25, 37D30 ; Secondary: 37C70, 37C75.

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