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

Abstract

Consider a topologically transitive countable Markov shift $\Sigma$ and a summable Markov potential $\varphi$ with finite Gurevich pressure and ${\mathrm{Var}}_1(\varphi )<\infty$. We prove 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 $t\varphi$. Besides that, we present examples where the limit at zero temperature exists for potentials satisfying more general conditions.

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Cited by in corpus (1)

• Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials