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HOMOCLINIC CLASSES AND FINITUDE OF ATTRACTORS FOR VECTOR FIELDS ON $n$-MANIFOLDS

Published online by Cambridge University Press:  24 March 2003

C. M. CARBALLO  and
C. A. MORALES
C. M. CARBALLO
Affiliation:
PUC-Rio, Dto. de Matemática, Rua Marquês de São Vicente, 225, 22453-900, Rio de Janeiro, RJ, [email protected]
C. A. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21945-970, Rio de Janeiro, RJ, [email protected]
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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 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.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 1 , January 2003 , pp. 85 - 91
Copyright
© The London Mathematical Society 2003

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Footnotes

The first author was supported by CNPq/Brazil. The second author was supported by FAPERJ/Brazil, CNPq, and PRONEX-Dyn. Sys./Brazil.