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 Ws (Λ). Our main result states that z ∈Ws (Λ) is accessible from M \ Ws (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 Ws (Λ).
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