  A homoclinic class of a vector field 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 C1 vector field on a closed n-manifold has
either infinitely many homoclinic classes or a finite collection of attractors
(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 C1 vector fields on a 5-manifold having finitely many attractors and
repellers but infinitely many homoclinic classes.
