Homoclinic Classes and Nitude of Attractors for Vector Elds on N-manifolds

A homoclinic class of a vector eld is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor (a repeller) is a transitive set to which converges every positive (negative) nearby orbit. We show that a generic C 1 vector eld on a closed n-manifold has either innnitely many homoclinic classes or a nite collection of attrac-tors (repellers) whose basins form an open-dense set. This result gives an approach to a conjecture by Palis. We also prove the existence of a locally residual subset of C 1 vector elds on a 5-manifold having nitely many attractors and repellers but innnitely many homoclinic classes.