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On the relation between Gibbs and $g$-measures

Published online by Cambridge University Press:  28 March 2018

STEVEN BERGHOUT Open the ORCID record for STEVEN BERGHOUT [Opens in a new window] ,
ROBERTO FERNÁNDEZ  and
EVGENY VERBITSKIY
STEVEN BERGHOUT
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands email [email protected]
ROBERTO FERNÁNDEZ
Affiliation:
Department of Mathematics, Utrecht University, PO Box 80010, 3508 TA Utrecht, The Netherlands email [email protected]
EVGENY VERBITSKIY
Affiliation:
Mathematical Institute, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands email [email protected] Department of Mathematics, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands email [email protected]
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Abstract

Thermodynamic formalism, the theory of equilibrium states, is studied both in dynamical systems and probability theory. Various closely related notions have been developed: e.g. Dobrushin–Lanford–Ruelle Gibbs, Bowen–Gibbs and $g$-measures. We discuss the relation between Gibbs and $g$-measures in a one-dimensional context. Often $g$-measures are also Gibbs, but recently an example to the contrary has been presented. In this paper we discuss exactly when a $g$-measure is Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3224 - 3249
Copyright
© Cambridge University Press, 2018 

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