A Criterion for Potentially Good Reduction in Non-archimedean Dynamics

TL;DR

This paper establishes a precise criterion based on fixed points and their preimages to determine when a non-archimedean dynamical system has potentially good reduction, aiding understanding of its stability.

Contribution

It provides a necessary and sufficient condition involving fixed points for assessing potentially good reduction in non-archimedean dynamics.

Findings

Criterion based solely on fixed points and preimages

Necessary and sufficient condition established

Applicable to polynomials and rational functions of degree ≥ 2

Abstract

Let K be a non-archimedean field, and let f in K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of f and their preimages, that determines whether or not the dynamical system f on P^1 has potentially good reduction.