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

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 $\text{F}$ be an almost-additive and summable potential with bounded variation potential. We prove that there exists an unique equilibrium state ${\mu }_{t\text{F}}$ for each $t>1$ and there exists an accumulation point ${\mu }_{\mathrm{\infty }}$ for the family $({\mu }_{t\text{F}}{)}_{t>1}$ as $t\to \mathrm{\infty }$. We also obtain that the Gurevich pressure $P_G(t\text{F})$ is $C^1$ on $(1,\mathrm{\infty })$ and the Kolmogorov-Sinai entropy $h({\mu }_{t\text{F}})$ is continuous at $(1,\mathrm{\infty })$. 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.