$p$-adic rational maps having empty Fatou set

TL;DR

This work demonstrates the existence of p-adic rational maps φ ∈ K(z) with empty Fatou set on the projective line over any finite extension K of \mathbb{Q}_p. The authors construct φ so that points outside the ring of integers map into the maximal ideal and φ acts on the integers as a small perturbation of a base map, making (\mathcal{O}_K, φ) dynamically conjugate to a full shift on p^f symbols; this conjugacy propagates to the global dynamics on P^1(K). They provide explicit families, such as φ(z) = \frac{z^{p^f}-z}{\pi+ z^{p^{mf}}- z^{p^{nf}}} (m>n) and φ(z) = \frac{z^{p}-z}{p+ z^{p^{m}}- z^{p^{n}}} (m>n), to realize empty Fatou sets. The results connect p-adic dynamics with symbolic dynamics via reduction modulo the uniformizer, establishing that the Julia set can be the entire space in this setting and highlighting the richness of Fatou/Julia phenomena in non-archimedean dynamics.

Abstract

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

Theorems & Definitions (13)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1: AK09, p.56
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof