A Variational Principle for the Pressure of Continuous Transformations

Walters, Peter

doi:10.2307/2373682

Cited by 322 publications

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“…If we denote the pressure of T by PT: CXX) -*■ R U {+°°} then this is equivalent to requiring h^XJ) + pi(0) = PT(<p) [12]. If the mapping u -*■ hJT) of M(T) to R is upper semicontinuous, then each 0 £ CXX) has equilibrium states.…”

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confidence: 99%

Ruelle’s operator theorem and 𝑔-measures

Walters¹

1975

Trans. Amer. Math. Soc.

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ABSTRACT. We use ^-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 J24 is a basic set of an Axiom A diffeomorphism / then, with respect to f\ns' each Holder continuous c4: J2S -> R has a unique equilibrium state and if f\cis is topologically mixing then with respect to the equilibrium state f\ns 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 CLX) 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 / G C(X) and cGR (here T denotes the one-sided shift). In §4 we extend these results to transformations more general than one-sided subshifts of finite type.

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confidence: 99%

Ruelle’s operator theorem and 𝑔-measures

Walters¹

1975

Trans. Amer. Math. Soc.

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“…Our general approach is motivated by thermodynamic formalism, in particular the theory of pressure and equilibrium states developed by Ruelle [48], [49] and Walters [54], [55]. After briefly recalling some notions from convex geometry in section 2, we develop some thermodynamic ideas in section 3.…”

Section: Consider a Continuous Mapmentioning

confidence: 99%

“…Consequently any ergodic measure supported on Y α will be directional, and in the fibre f −1 * ( ). Moreover, the variational principle (see Walters [54], [55]) means we can find an ergodic measure µ α , with supp(…”

Section: Theorem 5 Let X Be a Mixing Subshift Of Finite Type Let F mentioning

confidence: 99%

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Rotation, entropy, and equilibrium states

Jenkinson

2001

Trans. Amer. Math. Soc.

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Abstract. For a dynamical system (X, T ) and function f : X → R d we consider the corresponding generalised rotation set. This is the convex subset of R d consisting of all integrals of f with respect to T -invariant probability measures. We study the entropy H( ) of rotation vectors , and relate this to the directional entropy H( ) of Geller & Misiurewicz. For (X, T ) a mixing subshift of finite type, and f of summable variation, we prove that if the rotation set is strictly convex then the functions H and H are in fact one and the same. For those rotation sets which are not strictly convex we prove that H( ) and H( ) can differ only at non-exposed boundary points .

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“…Walters [2] generalized it to the general case and established the classical variational principle, which states that the topological pressure is the supremum of the measure-theoretic entropy together with the integral of the potential over all invariant measures. In the special case that the potential is zero, it reduces to the variational principle for topological entropy.…”

Section: Introductionmentioning

confidence: 99%