Ruelle's zeta function for non-Archimedean rational maps

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.

9 pages, 1 figure