Connected components of Berkovich fixed locus: Potential good reduction

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.