Symbolic extensions and smooth dynamical systems

Let f : X --> X be a homeomorphism of the compact metric space X. A symbolic extension of ( f, X) is a subshift on a finite alphabet ( g, Y) which has f as a topological factor. We show that a generic C-1 nonhyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r > 1, we exhibit a residual subset R of an open set U of C-r diffeomorphisms of a compact surface such that if f is an element of R, then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C-infinity diffeomorphism has symbolic extensions with the same entropy. We also show that C-r generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.