Skew Products on the Berkovich Projective Line

2023/10/26 by Richard A. P. Birkett, Birkett, Richard A. P.
Mathematics · #37P40 #37P50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2310.17628

openalex publication_date 2023/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we develop a dynamical theory for what shall be called a skew product on the Berkovich projective line, ϕ_*: ℙ¹_an(K) → ℙ¹_an(K) over a non-Archimedean field K. These functions are defined algebraically yet strictly generalise the notion of a rational map on ℙ¹_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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