Compressive Domains and a Bound for the Number of Components of the Fixed Locus of a Self-Map of the Berkovich Line

Abstract

We introduce the notion of a "compressive domain" for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. We prove that such a domain always contains a classical fixed point, and we leverage this fact to give a sharp upper bound for the number of connected components of the fixed locus of a rational function. We give a second proof for polynomial functions that uses a previously unpublished mass formula of Rivera-Letelier. Finally, we give an explicit formula for the crucial weight inside a compressive domain as a function of the number of classical fixed points and boundary points.