Stability of quasi-transversal bifurcation of vector fields on 3-manifolds

Abstract

Stability of generic arcs of hyperbolic vector fields having a bifurcation due to the creation of a quasi-transversal intersection orbit is studied under the assumption that The main novelty is the treatment that we give to the case where this quasi-transversal orbit is in the intersection of the unstable manifold of some orbit of a non-trivial basic set: we prove its stability using some special neighbourhood structure that resembles Thurston's `train tracks'. Following closely the ideas of Palis and Smale, our approach also provides a direct geometric proof of the stability of hyperbolic flows satisfying the transversality condition, a fact proved in all dimensions by Robinson. This original geometric approach has played a key role in the study of bifurcation and stability of parametrized families of vector fields as described by several authors.