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Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures

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Walters, Peter, 1943-. (2005) Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures. Journal of the London Mathematical Society, Vol.71 (No.2). pp. 379-396. ISSN 0024-6107

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Official URL: http://dx.doi.org/10.1112/S0024610704006076

Abstract

When T : X -> X is a one-sided topologically mixing subshift of finite type and {varphi} : X -> R is a continuous function, one can define the Ruelle operator L{varphi} : C(X) -> C(X) on the space C(X) of real-valued continuous functions on X. The dual operator Formula always has a probability measure {nu} as an eigenvector corresponding to a positive eigenvalue (Formula = {lambda}{nu} with {lambda} > 0). Necessary and sufficient conditions on such an eigenmeasure {nu} are obtained for {varphi} to belong to two important spaces of functions, W(X, T) and Bow (X, T). For example, {varphi} isin Bow(X, T) if and only if {nu} is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state µ{varphi} of {varphi} isin Bow(X, T) has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of g-measures to obtain results on the ‘reverse’ of a g-measure.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Bernoulli shifts, Transformations (Mathematics), Eigenvectors, Equilibrium, Ergodic theory
Journal or Publication Title:	Journal of the London Mathematical Society
Publisher:	Cambridge University Press
ISSN:	0024-6107
Date:	April 2005
Volume:	Vol.71
Number:	No.2
Page Range:	pp. 379-396
Identification Number:	10.1112/S0024610704006076
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/735