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

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 \cite{MaUr01} and \cite{Sar99} for the existence of equilibrium states without the BIP property, \cite{JMU05} for the existence of accumulation points in this case and, finally, we extend completely the result of \cite{Mor07} for the entropy zero temperature limit beyond the finitely primitive case.

Theorem 2 has been removed due to a gap in its final proof, and theorem 3 has been extended with a new proof. Other smaller fixes suggested by the referee have been included

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