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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. 379396. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S0024610704006076
Abstract
When T : X > X is a onesided 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 realvalued 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 measuretheoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a twosided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of gmeasures to obtain results on the ‘reverse’ of a gmeasure.
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:  00246107 
Official Date:  April 2005 
Volume:  Vol.71 
Number:  No.2 
Page Range:  pp. 379396 
Identification Number:  10.1112/S0024610704006076 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. T. Bousch, ‘La condition de Walters’, Ann. Sci. ´ Ecole Norm. Sup. 34 (2001) 287–311. 
URI:  http://wrap.warwick.ac.uk/id/eprint/735 
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