Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields

Abstract

We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincaré flow and a generalization of Mañé-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities.

References in corpus (1)

• Morse-Smale systems and horseshoes for three dimensional singular flows

Cited by in corpus (6)

• On the centralizer of vector fields: criteria of triviality and genericity results
• Symmetries of vector fields: the diffeomorphism centralizer
• Star flows with singularities of different indices
• Robust transitivity of singular hyperbolic attractors
• On the growth rate of periodic orbits for vector fields
• On the volume of sectional-hyperbolic sets