$p$-adic rational maps having empty Fatou set

Abstract

On any finite algebraic extension $K$ of the field $\Q_p$ of $p$-adic numbers, there exist rational maps $\varphi \in K(z)$ such that dynamical system $({\mathbb{P}}^1(K),\varphi )$ has empty Fatou set, i.e. the iteration family $\{{\varphi }^n:n\ge 0\}$ is nowhere equicontinuous.