Homoclinic classes and finitude of attractors for vector fields on n-manifolds

Details Homoclinic classes and finitude of attractors for vector fields on n-manifolds Homoclinic classes and finitude of attractors for vector fields on n-manifolds

2001-05-17

arxiv.org/abs/math/0105143v1

Mathematics

Dynamical Systems

12 pages, 3 figures

Scientific paper

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.

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