Singular-Hyperbolic Connecting Lemma

Abstract

Bautista and Morales (Ergod Theory Dyn Syst 30(2):339–359, 2010), present the sectional-Anosov connecting lemma as a property of a singular-hyperbolic attracting set $\Lambda$ on a compact three-dimensional manifold M; this property says that given any two points p and q in $\Lambda$, such that for every $\epsilon >0$, there is a trajectory from a point $\epsilon$-close to p to a point $\epsilon$-close to q, and p has non-singular $\alpha$-limit set, then there is a point in M whose $\alpha$-limit is that of p and whose $\omega$-limit is either a singularity or that of q. In this paper, we prove a generalization of this result, for singular-hyperbolic sets that contain the unstable manifolds of their hyperbolic subsets, although these are not necessarily attracting sets, also, we extend the result to compact manifolds of dimension greater or equal to 3 when the singular-hyperbolic set is of codimension one.