Periodic Orbits and Homoclinic Loops for Surface Homeomorphisms

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 fn whose index is ρ.