Connected components of Berkovich fixed locus: Potential good reduction
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912816279060480 |
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| author | Patra, Niladri |
| author_facet | Patra, Niladri |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connected components of Berkovich fixed locus: Potential good reduction Patra, Niladri Dynamical Systems Algebraic Geometry Number Theory 37P50 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. |
| title | Connected components of Berkovich fixed locus: Potential good reduction |
| topic | Dynamical Systems Algebraic Geometry Number Theory 37P50 |
| url | https://arxiv.org/abs/2508.07156 |