Dynamics Forced by Surface Trellises

Given a saddle fixed point of a surface diffeomorphism, its stable and unstable curves W S and W U often form a homoclinic tangle. Given such a tangle, we use topological methods to find periodic points of the diffeomorphism, using only a subset of the tangle with finitely many points of intersection, which we call a trellis. We typically obtain exponential growth of periodic orbits, symbolic dynamics and a strictly positive lower bound for topological entropy. For a simple example occurring in the Hénon family, we show that the topological entropy is at least 0.527.