Wandering domains for non-archimedean quadratic rational functions

TL;DR

This paper demonstrates the existence of wandering Fatou components for a specific class of non-archimedean quadratic rational functions over algebraically closed fields with residual characteristic 2.

Contribution

It constructs explicit examples of quadratic rational functions with wandering Fatou components in a non-archimedean setting, expanding understanding of dynamical behaviors in such fields.

Findings

Existence of wandering Fatou components for certain rational functions

Explicit construction of functions with wandering domains

Analysis within non-archimedean fields of residual characteristic 2

Abstract

Let $\mathbb{C}_K$ be a complete and algebraic closed non-archimedean field with residual characteristic $2$. In this paper we prove that there exist $a,b\in\mathbb{C}_K$ such that the rational function $R(z)=\frac{z^2-z}{bz-\frac{1}{a}}$ has wandering components in its Fatou set.