Maximal transitive sets with singularities for genericC 1 vector fields

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Abstract

A transitive set Λ of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC 1 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 genericC 1 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.