Topological entropy of homoclinic closures

Abstract:

In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in $n$-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen’s one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.