Connected components of Berkovich fixed locus: Potential good reduction

TL;DR

The paper analyzes the fixed-locus topology on the Berkovich line $P^{1,an}$ for degree $d ≥ 2$ rational maps with potential good reduction. It shows this topology is governed by the reduction at the unique type II totally ramified fixed point via the residual map, and it proves a precise component-count formula. It gives equivalent criteria for when the fixed locus is connected (no fixed critical point in the residual map at the Gauss point) and when it is finite (every fixed point of the residual map is critical), along with a sharp bound $d+2$ on the number of components. Together, these results clarify how ramification interacts with fixed points in non-Archimedean dynamics and provide practical tools for analyzing Berkovich periodic loci.

Abstract

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.

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Proposition 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 2.5