Accessibility and hyperbolicity

We examine conditions under which a point in the stable set of a hyperbolic invariant set for a C¹ surface diffeomorphism is accessible via a path from the complement of the stable set. Let M be a surface, and let Λ be a compact saturated hyperbolic locally stably closed invariant set possessing a local product structure. Denote the stable set of Λ by W^s (Λ). Our main result states that z ∈W^s (Λ) is accessible from M \ W^s (Λ) if and only if z lies on the stable manifold of a periodic point p, and there is a branch of a local unstable manifold of p disjoint from W^s (Λ).