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

2025/05/22 by Jie Cao, Cao, Jie
Engineering · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.16729

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 μ_t\F for each t>1 and there exists an accumulation point μ_∞ for the family (μ_t\F)_t>1 as t→∞. We also obtain that the Gurevich pressure P_G(t\F) is C¹ on (1,∞) and the Kolmogorov-Sinai entropy h(μ_t\F) is continuous at (1,∞). 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.

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