Periodic orbits and homoclinic loops for surface homeomorphisms.

Abstract

Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f, admitting a homoclinic point p ′ . Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to p ′ , lying respectively in the stable and unstable curves at p. It is shown that f |V has fixed point index ρ ∈ {1,2} where ρ depends only on the geometry of V near p. More generally, for every n ≥ 1, the union of the n-periodic orbits in V is a block of fixed points for f n whose index is ρ.