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arXiv · 2503.10018

Zeta function and entropy for non-archimedean subhyperbolic dynamics

Liang-Chung Hsia·Hongming Nie·Chenxi Wu
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Abstract

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.

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Liang-Chung Hsia, Hongming Nie, Chenxi Wu. 2025-03-13. Zeta function and entropy for non-archimedean subhyperbolic dynamics. https://arxiv.org/abs/2503.10018

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