SRB measures for Axiom A endomorphisms

Abstract

Let A be a basic set of an Axiom A endomorphism f on a n-dimensional compact Riemannian manifold. In this paper, we provide equivalent conditions for the existence of an SRB measure on A. In particular, we show that under the assumption that the closure of the postcritical set of f is disjoint from A, the existence of an SRB measure is equivalent to the condition that the stable set of A has packing dimension n. Our approach uses the theory of thermodynamic formalism. In particular, we consider the topological pressure on the inverse limit space.