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Transactions of the American Mathematical Society

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Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts


Authors: Ricardo Freire and Victor Vargas
Journal: Trans. Amer. Math. Soc. 370 (2018), 8451-8465
MSC (2010): Primary 28Dxx, 37Axx
DOI: https://doi.org/10.1090/tran/7291
Published electronically: July 12, 2018
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Abstract: Consider a topologically transitive countable Markov shift and, let $ f$ be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state $ \mu _{tf}$ for each $ t > 1$ and that there exists accumulation points for the family $ (\mu _{tf})_{t>1}$ as $ t \to \infty $. We also prove that the Kolmogorov-Sinai entropy is continuous at $ \infty $ with respect to the parameter $ t$, that is, $ \lim _{t \to \infty } h(\mu _{tf})=h(\mu _{\infty })$, where $ \mu _{\infty }$ is an accumulation point of the family $ (\mu _{tf})_{t>1}$. These results do not depend on the existence of Gibbs measures and, therefore, they extend results of [Israel J. Math. 125 (2001), pp. 93-130] and [Ergodic Theory Dynam. Systems 19 (1999), pp. 1565-1593] for the existence of equilibrium states without the big images and preimages (BIP) property, [J. Stat. Phys. 119 (2005), pp. 765-776] for the existence of accumulation points in this case and, finally, we extend completely the result of [J. Stat. Phys. 126 (2007), pp. 315-324] for the entropy zero temperature limit beyond the finitely primitive case.


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Additional Information

Ricardo Freire
Affiliation: Department of Mathematics, IME-USP, Rua do Matão, 1010, São Paulo, Brazil
Email: rfreire@usp.br

Victor Vargas
Affiliation: Department of Mathematics, IME-USP, Rua do Matão, 1010, São Paulo, Brazil
Address at time of publication: Faculty of Education, Antonio Nariño University, Cl. 22 Sur 12D-81, Bogotá, Colombia
Email: vavargascu@gmail.com

DOI: https://doi.org/10.1090/tran/7291
Keywords: Equilibrium states, Gibbs measures, summable potentials, Markov shifts, zero temperature limit.
Received by editor(s): November 24, 2015
Received by editor(s) in revised form: March 26, 2017
Published electronically: July 12, 2018
Additional Notes: The first author was supported by FAPESP process 2011/16265-8.
The second author was supported by CAPES. Parts of these results were in the author’s Ph.D. thesis.
Article copyright: © Copyright 2018 American Mathematical Society

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