On computability of equilibrium states
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866918130985467904 |
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| author | Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei |
| author_facet | Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei |
| contents | Equilibrium states are natural dynamical analogues of Gibbs states in thermodynamic formalism. This paper investigates their computability within the framework of Computable Analysis. We show that the unique equilibrium state for a computable, open, topologically exact, distance-expanding map $T\colon X\rightarrow X$ and a computable Hölder continuous potential $φ\colon X\rightarrow\mathbb{R}$ is always computable. As an application, we establish the computability of equilibrium states for computable hyperbolic rational maps and their respective geometric potentials. Moreover, we develop a constructive method to exhibit the non-uniqueness of equilibrium states for some dynamical systems. We also present some computable dynamical systems whose equilibrium states are all non-computable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09374 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On computability of equilibrium states Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei Dynamical Systems Logic Primary: 03D78, Secondary: 37D35, 37D20, 37F15 Equilibrium states are natural dynamical analogues of Gibbs states in thermodynamic formalism. This paper investigates their computability within the framework of Computable Analysis. We show that the unique equilibrium state for a computable, open, topologically exact, distance-expanding map $T\colon X\rightarrow X$ and a computable Hölder continuous potential $φ\colon X\rightarrow\mathbb{R}$ is always computable. As an application, we establish the computability of equilibrium states for computable hyperbolic rational maps and their respective geometric potentials. Moreover, we develop a constructive method to exhibit the non-uniqueness of equilibrium states for some dynamical systems. We also present some computable dynamical systems whose equilibrium states are all non-computable. |
| title | On computability of equilibrium states |
| topic | Dynamical Systems Logic Primary: 03D78, Secondary: 37D35, 37D20, 37F15 |
| url | https://arxiv.org/abs/2311.09374 |