Ruelle's zeta function for non-Archimedean rational maps

TL;DR

This paper investigates Ruelle's zeta functions for non-Archimedean rational maps over p-adic fields, establishing their meromorphic nature and linking transfer operators to Julia set geometry.

Contribution

It introduces the study of Ruelle's zeta functions in the non-Archimedean setting and proves their meromorphicity for subhyperbolic rational maps over p-adic fields.

Findings

Ruelle's zeta functions are meromorphic on a2_p.

Transfer operators reveal Julia set structures.

A Levin-Sodin-Yuditski type identity is established.

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.