Expanding property and statistical laws for $p$-adic subhyperbolic rational maps

Abstract

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $\phi\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $\phi$ is eventually periodic. We show that subhyperbolic maps in $K(z)$ exhibit expanding property with respect to some (singular) metric. As an application, under a mild assumption, we establish several statistical laws for such maps in $K(z)$ with compact $\mathbb{C}_p$-Julia sets.