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

C. M. Carballo·C. A. Morales
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Abstract

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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C. M. Carballo, C. A. Morales. 2001-05-17. Homoclinic classes and finitude of attractors for vector fields on n-manifolds. https://arxiv.org/abs/math/0105143

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