Skew Products on the Berkovich Projective Line

Abstract

In this article, we develop a dynamical theory for what shall be called a skew product on the Berkovich projective line, ${\varphi }_{\ast }:{\mathbb{P}}_{\text{an}}^1(K)\to {\mathbb{P}}_{\text{an}}^1(K)$ over a non-Archimedean field $K$. These functions are defined algebraically yet strictly generalise the notion of a rational map on ${\mathbb{P}}_{\text{an}}^1$. We describe the analytical, algebraic, and dynamical properties of skew products, including a study of periodic points, and a Fatou/Julia dichotomy. The article culminates with the classification of the connected components of the Fatou set.

78 pages. The results in this article constitute chapter 3 of the author's PhD thesis. We acknowledge that H. Nie and S. Zhao have developed an alternative approach to these problems and have an independent proof of the classification of Fatou components