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arXiv · 2310.17628

Skew Products on the Berkovich Projective Line

Richard A. P. Birkett
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Abstract

In this article, we develop a dynamical theory for what shall be called a skew product on the Berkovich projective line, $\phi_*: \mathbb{P}^1_{\text{an}}(K) \to \mathbb{P}^1_{\text{an}}(K)$ over a non-Archimedean field $K$. These functions are defined algebraically yet strictly generalise the notion of a rational map on $\mathbb{P}^1_{\text{an}}$. 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.

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Richard A. P. Birkett. 2023-10-26. Skew Products on the Berkovich Projective Line. https://arxiv.org/abs/2310.17628

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