A spectral decomposition for singular-hyperbolic sets

Carlos A. Morales and Maria José Pacífico

Vol. 229 (2007), No. 1, 223–232

DOI: 10.2140/pjm.2007.229.223

Abstract

We extend the Spectral Decomposition Theorem for hyperbolic sets to singular-hyperbolic sets on 3-manifolds. We prove that an attracting singular-hyperbolic set with dense periodic orbits and a unique equilibrium of a C^r vector field, where r ≥ 1, is a finite union of transitive sets; the union is disjoint or the set contains finitely many distinct homoclinic classes. If the vector field is C^r-generic, the union is in fact disjoint.

Keywords

transitive set, singular-hyperbolic set, homoclinic class, spectral decomposition

Mathematical Subject Classification 2000

Primary: 37D30

Secondary: 37D50

Milestones

Received: 10 June 2005

Accepted: 29 November 2005

Published: 1 January 2007