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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.16729 |
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| _version_ | 1866912459247321088 |
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| author | Cao, Jie |
| author_facet | Cao, Jie |
| contents | 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 $μ_{\infty}$ for the family $(μ_{t\F})_{t>1}$ 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 $h(μ_{t\F})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equilibrium States, Zero Temperature Limits and Entropy Continuity for Almost-Additive Potentials Cao, Jie Dynamical Systems 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 $μ_{\infty}$ for the family $(μ_{t\F})_{t>1}$ 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 $h(μ_{t\F})$ 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. |
| title | Equilibrium States, Zero Temperature Limits and Entropy Continuity for Almost-Additive Potentials |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2505.16729 |