Structure of the Components of the Fixed Locus of a Self-Map of the Berkovich Line

TL;DR

This paper analyzes the fixed points of rational functions acting on the Berkovich projective line over nonarchimedean fields, providing bounds on their connected components and detailing their local and global structure.

Contribution

It offers a detailed description of the fixed locus structure, including sharp bounds on the number of components in certain residue characteristic cases.

Findings

Bound on the number of connected components of the fixed locus

Description of local and global structure of fixed points

Results are sharp in large or zero residue characteristic cases

Abstract

We describe the local and global structure of the fixed locus for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. This includes a bound for the number of connected components that is sharp when the residue characteristic is large or zero. The case of small nonzero residue characteristic will be treated in a subsequent article.