The Lipschitz Constant of a Nonarchimedean Rational Function

TL;DR

This paper establishes explicit bounds for the Lipschitz constants of nonarchimedean rational functions on the Berkovich projective line and classical points, enhancing understanding of their metric behavior in nonarchimedean geometry.

Contribution

It provides the first explicit bounds for Lipschitz constants of nonarchimedean rational functions in the Berkovich setting, relating to Favre/Rivera-Letelier and spherical metrics.

Findings

Explicit bounds for Lipschitz constants on Berkovich line

Bounds for classical points in projective line

Improved understanding of nonarchimedean dynamics

Abstract

Let K be a complete, algebraically closed nonarchimedean valued field, and let f(z) be a non-constant rational function in K(z). We provide explicit bounds for the Lipschitz constant of f(z) acting on the Berkovich projective line, relative to the Favre/Rivera-Letelier d(x,y)-metric, and for the Lipschitz constant of f(z) acting on classical points in the projective line, relative to the spherical metric.