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Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures
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Walters, Peter (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. doi:10.1112/S0024610704006076 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | April 2005 | ||||
Dates: |
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Volume: | Vol.71 | ||||
Number: | No.2 | ||||
Page Range: | pp. 379-396 | ||||
DOI: | 10.1112/S0024610704006076 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Open Access (Creative Commons) |
Data sourced from Thomson Reuters' Web of Knowledge
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