Homoclinic Bifurcations and Uniform Hyperbolicity for Three-dimensional Flows
Federico Rodriguez-Hertz | Arroyo, Aubin
homoclinic bifurcations dominated splitting flow singular cycle
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 an analogous statement of a conjecture of Palis for diffeomorphisms in the context of $C^1$-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.