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• W2024525038 abstract "We use g-measures to give a proof of a convergence theorem of Ruelle. The method of proof is used to gain information about the ergodic properties of equilibrium states for subshifts of finite type. 0. In 1968 D. Ruelle proved a convergence theorem (Theorem 3 of [8]) which he used to obtain a large class of interactions, for an infinite one-dimensional classical lattice gas, which have no phase transitions. The equilibrium state of such a system was shown to have strong ergodic properties. In 1973 R. Bowen remarked that Ruelle's proof could be extended to show the convergence of the powers of a certain operator acting on the space of all real-valued continuous functions on a one-sided shift space [2]. In the paper [2] he used this result and the theory of Markov partitions, due to Ya. G. Sinai and himself, to show that if Q5 is a basic set of an Axiom A diffeomorphism f then, with respect to f In., each Holder continuous 0: Q5 R has a unique equilibrium state and if f In is topologically mixing then with respect to the equilibrium state f In,is a Bernoulli shift. This was done by using Markov partitions to move the problem to one about a two-sided subshift of finite type ([1], [10]). It can then be reduced to a problem about a one-sided subshift of finite type ([2], [11, p. 28]) and then Ruelle's theorem can be used. In this paper we connect these results with the idea of a g-measure, studied by M. Keane [5]. We shall give a proof of Ruelle's operator theorem using the notion of g-measure. The structure of this proof allows us to deduce elementary proofs of several results about equilibrium states. There is a dense subset V of C(X) whose members each have a unique equilibrium state, the natural extension of the one-sided shift with respect to these measures are Bernoulli shifts, and two members of V have the same equilibrium state if and only if they differ by a function of the form f ? T f + c where f E C(X) and c E R (here T denotes the one-sided shift). In ?4 we extend these results to transformations more general than one-sided subshifts of finite type. Received by the editors December 10, 1974. AMS (MOS) subject classifications (1970). Primary 28A65, 58F15. Copyright" @default.
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• W2024525038 date "1975-01-01" @default.
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• W2024525038 title "Ruelle’s operator theorem and $g$-measures" @default.
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