Equilibrium States, Zero Temperature Limits and Entropy Continuity for Almost-Additive Potentials

TL;DR

This paper investigates equilibrium states for almost-additive potentials on non-compact topologically mixing countable Markov shifts, establishing existence, uniqueness, and properties of these states, including zero temperature limits and entropy continuity.

Contribution

It extends previous results on equilibrium states, zero temperature limits, and entropy continuity to broader classes of non-compact Markov shifts with almost-additive potentials.

Findings

Unique equilibrium states exist for each t>1.

Gurevich pressure is C^1 on (1,∞).

Entropy is continuous at infinity.

Abstract

This paper is devoted to study the equilibrium states for almost-additive potentials defined over topologically mixing countable Markov shifts (that is a non-compact space) without the big images and preimages (BIP) property. Let $\F$ be an almost-additive and summable potential with bounded variation potential. We prove that there exists an unique equilibrium state $\mu_{t\F}$ for each $t>1$ and there exists an accumulation point $\mu_{\infty}$ for the family $(\mu_{t\F})_{t>1}$ as $t\to\infty$. We also obtain that the Gurevich pressure $P_{G}(t\F)$ is $C^1$ on $(1,\infty)$ and the Kolmogorov-Sinai entropy $h(\mu_{t\F})$ is continuous at $(1,\infty)$. As two applications, we extend completely the results for the zero temperature limit [J. Stat. Phys. ,155 (2014),pp. 23-46] and entropy continuity at infinity [J. Stat. Phys., 126 (2007),pp. 315-324] beyond the finitely primitive case. We also extend the result [Trans. Amer. Math. Soc., 370 (2018), pp. 8451-8465] for almost-additive potentials.