Zeta function and entropy for non-archimedean subhyperbolic dynamics
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arXiv
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| Format: | Preprint |
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2025
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| author | Hsia, Liang-Chung Nie, Hongming Wu, Chenxi |
| author_facet | Hsia, Liang-Chung Nie, Hongming Wu, Chenxi |
| contents | Let $K$ be a complete non-archimedean field of characteristic $0$ equipped with a discrete valuation. We establish the rationality of the Artin-Mazur zeta function on the Julia set for any subhyperbolic rational map defined over $K$ with a compact Julia set. Furthermore, we conclude that the topological entropy on the Julia set of such a map is given by the logarithm of a weak Perron number. Conversely, we construct a (sub)hyperbolic rational map defined over $K$ with compact Julia set whose topological entropy on the Julia set equals the logarithm of a given weak Perron number. This extends Thurston's work on the entropy for postcritically finite interval self-maps %of the unit interval to the non-archimedean setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeta function and entropy for non-archimedean subhyperbolic dynamics Hsia, Liang-Chung Nie, Hongming Wu, Chenxi Dynamical Systems Number Theory 37P05, 37P10, 11S82, 37E25, 37C35, 37B40 Let $K$ be a complete non-archimedean field of characteristic $0$ equipped with a discrete valuation. We establish the rationality of the Artin-Mazur zeta function on the Julia set for any subhyperbolic rational map defined over $K$ with a compact Julia set. Furthermore, we conclude that the topological entropy on the Julia set of such a map is given by the logarithm of a weak Perron number. Conversely, we construct a (sub)hyperbolic rational map defined over $K$ with compact Julia set whose topological entropy on the Julia set equals the logarithm of a given weak Perron number. This extends Thurston's work on the entropy for postcritically finite interval self-maps %of the unit interval to the non-archimedean setting. |
| title | Zeta function and entropy for non-archimedean subhyperbolic dynamics |
| topic | Dynamical Systems Number Theory 37P05, 37P10, 11S82, 37E25, 37C35, 37B40 |
| url | https://arxiv.org/abs/2503.10018 |