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On balanced subshifts and the existence of invariant Gibbs measures Jung, Uijin
Description
It is well known that if $f$ is a Holder continuous function from a mixing shift of finite type $X$ to $\mathbb R$, then there exists a unique equilibrium state which is an invariant Gibbs measure having $f$ as a potential function. This result has been generalized to wider classes, such as when $X$ is a subshift with the specification property and $f$ is a function in the Bowen class. Recently Baker and Ghenciu showed that there exists a (non-invariant) Gibbs measures for the zero potential if and only if $X$ is (right-)balanced. We extend this result and show that a necessary and sufficient condition for the existence of invariant Gibbs measures on $X$ for the potential $0$ is the bi-balanced condition for $X$. We define a new condition, called $f$-balanced condition for the pair $(X,f)$ and present a similar result for the existence of Gibbs measure with respect to $f$. Using this result, we construct a class of shift spaces which have a Gibbs measure but do not have invariant Gibbs measures for the potential $0$, or equivalently, which are one-sided balanced but not bi-balanced, answering a question raised by Baker and Ghenciu.
Item Metadata
Title |
On balanced subshifts and the existence of invariant Gibbs measures
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-05-14T11:30
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Description |
It is well known that if $f$ is a Holder continuous function from a mixing shift of finite type $X$ to $\mathbb R$, then there exists a unique equilibrium state which is an invariant Gibbs measure having $f$ as a potential function. This result has been generalized to wider classes, such as when $X$ is a subshift with the specification property and $f$ is a function in the Bowen class. Recently Baker and Ghenciu showed that there exists a (non-invariant) Gibbs measures for the zero potential if and only if $X$ is (right-)balanced. We extend this result and show that a necessary and sufficient condition for the existence of invariant Gibbs measures on $X$ for the potential $0$ is the bi-balanced condition for $X$. We define a new condition, called $f$-balanced condition for the pair $(X,f)$ and present a similar result for the existence of Gibbs measure with respect to $f$. Using this result, we construct a class of shift spaces which have a Gibbs measure but do not have invariant Gibbs measures for the potential $0$, or equivalently, which are one-sided balanced but not bi-balanced, answering a question raised by Baker and Ghenciu. |
Extent |
46.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Ajou University
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Series | |
Date Available |
2019-11-11
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0385158
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International