Symbolic extensions and smooth dynamical systems

Let f:XX 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 C¹ non-hyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r>1, we exhibit a residual subset of an open set of C^r diffeomorphisms of a compact surface such that if , then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C 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.