A spectral decomposition for singular-hyperbolic sets

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 Cr 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 Cr-generic, the union is in fact disjoint.