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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.07156 |
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Table of Contents:
- Let $\mathbbm{P}^{1,an}$ be the Berkovich projective line over a complete, algebraically closed, non-Archimedean field. Let $ϕ$ be a degree $\geq 2$ rational map with potential good reduction, acting on $\mathbbm{P}^{1,an}$. In this article, we study the topology of the fixed locus of $ϕ$. we show that the reduction of $ϕ$ at its type~II totally ramified fixed point dictates the topological structure of the fixed locus of $ϕ$. We give an easily verifiable equivalent criterion for the fixed locus of $ϕ$ to be connected as well as an equivalent criterion for the fixed locus of $ϕ$ to be finite. Moreover, we provide a sharp upper bound for the number of connected components of the fixed locus of a rational map with potential good reduction.