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arXiv · 2508.19374

Ruelle's zeta function for non-Archimedean rational maps

Yunping Jiang·Chenxi Wu
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Abstract

We studied the transfer operators defined over $\mathbb{C}_p$-valued analytic functions for subhyperbolic rational maps on $\mathbb{Q}_p$, and showed that the corresponding Ruelle's zeta functions are meromorphic on $\mathbb{C}_p$. We also used $\mathbb{R}$-valued transfer operators to study the shape of the corresponding Julia sets, and proved a Levin-Sodin-Yuditski type identity for general rational maps on $\mathbb{C}_p$. In all the results above, $\mathbb{Q}_p$ can be replaced with any non-Archimedean local field with characteristic $0$, and $\mathbb{C}_p$ the metric completion of its algebraic closure.

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Yunping Jiang, Chenxi Wu. 2025-08-26. Ruelle's zeta function for non-Archimedean rational maps. https://arxiv.org/abs/2508.19374

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