Homoclinic Classes and Finitude of Attractors For Vector Fields On n ‐Manifolds

A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor is a transitive set to which every positive nearby orbit converges; likewise, every negative nearby orbit converges to a repeller . It is shown in this paper that a generic C 1 vector field on a closed n ‐manifold has either infinitely many homoclinic classes, or a finite collection of attractors (or, respectively, repellers) with basins that form an open‐dense set. This result gives an approach to use in proving a conjecture by Palis. A proof is also given of the existence of a locally residual subset of C 1 vector fields on a 5‐manifold having finitely many attractors and repellers, but infinitely many homoclinic classes.