STABILITY OF QUASI-TRANSVERSAL BIFURCATION OF VECTOR-FIELDS ON 3-MANIFOLDS

Stability of generic arcs of hyperbolic vector fields having a bifurca tion due to the creation of a quasi-transversal intersection orbit is studied under the assumption that dim M = 3. The main novelty is the t reatment that we give to the case where this quasi-transversal orbit i s in the intersection of the unstable manifold of some orbit of a non- trivial basic set: we prove its stability using some special neighbour hood structure that resembles Thurston's 'train tracks'. Following clo sely the ideas of Palls and Smale, our approach also provides a direct geometric proof of the stability of hyperbolic Bows satisfying the tr ansversality condition, a fact proved in all dimensions by Robinson. T his 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.