Homoclinic Bifurcations and Uniform Hyperbolicity for three-dimensional flows
homoclinic biffurcations dominated splitting flow
In this paper we prove that any $C^1$ vector field defined on a three dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves a statement of a conjecture of Palis in the context of flows in the $C^1$ topology. For that, we rely on the notion of dominated splitting for the associated linear Poincare flow.