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

TL;DR

This paper investigates equilibrium states and entropy behavior at zero temperature for topologically transitive countable Markov shifts, extending existing results to broader classes without Gibbs measures and finite primitivity.

Contribution

It proves existence and accumulation of equilibrium states for all t>1 and shows entropy continuity at infinity, extending prior results to non-BIP and non-finitely primitive cases.

Findings

Existence of equilibrium states for all t>1.

Accumulation points of equilibrium states as t approaches infinity.

Continuity of entropy at zero temperature limit.

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.