SRB measures for Axiom A endomorphisms

Let $\Lambda$ 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 $\Lambda$. In particular, we show that under the assumption that the closure of the postcritical set of $f$ is disjoint from $\Lambda$, the existence of an SRB measure is equivalent to the condition that the stable set of $\Lambda$ has packing dimension $n$. Our approach uses the theory of thermodynamic formalism. In particular, we consider the topological pressure on the inverse limit space.

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Published 1 January 2004