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Existence of the zero-temperature limit of equilibrium states on topologically transitive countable Markov shifts
Published online by Cambridge University Press: 04 October 2022
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.
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- © The Author(s), 2022. Published by Cambridge University Press