Connected components of Berkovich fixed locus: Potential good reduction

TL;DR

This paper investigates the topological structure of fixed loci of rational maps with potential good reduction on the Berkovich projective line, providing criteria for connectedness and finiteness, and establishing an upper bound on the number of components.

Contribution

It offers new criteria for the connectedness and finiteness of fixed loci and establishes a sharp upper bound for their number in the context of potential good reduction.

Findings

Reduction at type II fixed point determines fixed locus topology

Criteria for connectedness and finiteness of fixed locus

Sharp upper bound for number of fixed components

Abstract

Let $\mathbbm{P}^{1,an}$ be the Berkovich projective line over a complete, algebraically closed, non-Archimedean field. Let $\phi$ 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 $\phi$. we show that the reduction of $\phi$ at its type~II totally ramified fixed point dictates the topological structure of the fixed locus of $\phi$. We give an easily verifiable equivalent criterion for the fixed locus of $\phi$ to be connected as well as an equivalent criterion for the fixed locus of $\phi$ 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.