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Existence of the zero-temperature limit of equilibrium states on topologically transitive countable Markov shifts

Part of: Dynamical systems with hyperbolic behavior Ergodic theory Markov processes

Published online by Cambridge University Press:  04 October 2022

ELMER BELTRÁN Open the ORCID record for ELMER BELTRÁN [Opens in a new window] ,
JORGE LITTIN Open the ORCID record for JORGE LITTIN [Opens in a new window] ,
CESAR MALDONADO Open the ORCID record for CESAR MALDONADO [Opens in a new window]  and
VICTOR VARGAS Open the ORCID record for VICTOR VARGAS [Opens in a new window]
ELMER BELTRÁN
Affiliation:
Departamento de Matemáticas, Universidad Católica del Norte, Avenida Angamos 0610, Antofagasta, Chile (e-mail: rusbert.unt@gmail.com)
JORGE LITTIN*
Affiliation:
Departamento de Matemáticas, Universidad Católica del Norte, Avenida Angamos 0610, Antofagasta, Chile (e-mail: rusbert.unt@gmail.com)
CESAR MALDONADO
Affiliation:
IPICYT, División de Control y Sistemas Dinámicos, Camino a la Presa San José 2055, Lomas 4a. sección, San Luis Potosí, México (e-mail: cesar.maldonado@ipicyt.edu.mx)
VICTOR VARGAS
Affiliation:
Center for Mathematics of the University of Porto, Rua do Campo Alegre 687, Porto, Portugal (e-mail: vavargascu@gmail.com)
*
e-mail: jlittin@ucn.cl
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Abstract

Consider a topologically transitive countable Markov shift $\Sigma $ and a summable locally constant potential $\phi $ with finite Gurevich pressure and $\mathrm {Var}_1(\phi ) < \infty $. We prove the existence of the limit $\lim _{t \to \infty } \mu _t$ in the weak$^\star $ topology, where $\mu _t$ is the unique equilibrium state associated to the potential $t\phi $. In addition, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.

Keywords

countable Markov shiftequilibrium statemaximizing measurerenewal shiftstationary Markov measuretopologically transitive

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 37A60: Dynamical systems in statistical mechanics 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 10 , October 2023 , pp. 3231 - 3254
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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