Maximal transitive sets with singularities for genericC 1 vector fields

A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC¹ generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC¹ vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.