2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61486A 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.12 pages, 3 figuresDynamical Systems37C20 (Primary), 37C29 (Secondary)text