A sectional-Anosov connecting lemma

Abstract

AbstractThe Anosov flows on compact manifolds M satisfy the following property: if p,q are points such that for all positive ϵ there is a trajectory from a point ϵ-close to p to a point ϵ-close to q, then there is a point whose α-limit set is that of p and whose ω-limit set is that of q. Here we give a version of this property for sectional-Anosov flows, namely, vector fields inwardly transverse to the boundary whose maximal invariant set is sectional-hyperbolic. Indeed, if in addition M is three-dimensional and p has non-singular α-limit set, then there is a point whose α-limit set is that of p and whose ω-limit set is either a singularity or that of q.