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 $\F$ 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.