Statistical mechanics on a compact set with $Z^\nu $ action satisfying expansiveness and specification

Abstract

We consider a compact set Q with a homeomorphism (or more generally a Z' action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on ??(s)—the real continuous functions on ft. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated. 0. Introduction. Invariant measures for an Anosov diffeomorphism have been studied by Sinai (16), (17). More generally, Bowen (2), (3) has considered invariant measures on basic sets for an Axiom A diffeomorphism. The problems encoun- tered are strongly reminiscent of those of statistical mechanics (for a classical lattice system—see (14, Chapter 7)). In fact Sinai (18) has explicitly used techniques of statistical mechanics to show that an Anosov diffeomorphism does not in general have a smooth invariant measure. In this paper, we rewrite a part of the general theory of statistical mechanics for the case of a compact set s satisfying expansiveness and the specification property of Bowen (2). Instead of a Z action we consider a Z' action as is usual in lattice statistical mechanics, where Q = Fz' (F: a finite set). This rewriting gives a more general and intrinsic formulation of (part of) statistical mechanics; it presents a number of technical problems, but the basic ideas are contained in the papers of Gallavotti, Lanford, Miracle, Robinson, and Ruelle (7), (11), (12), (13), etc. The ideas of Bowen (2) and Goodwyn (8) on the relation between topological and measure-theoretical entropy are also used. We describe now some of our results in the case of a homeomorphism F of a metrizable compact set n satisfying expansiveness and specification (see §1). Let na = {x G n: Tx = {x}}, and let <3(s) be the Banach space of real continuous functions on n. The pressure F is a continuous convex function on u(fi) defined by