Rational Functions with a Unique Critical Point

TL;DR

This paper characterizes rational functions with a single critical point over algebraically closed fields of positive characteristic, exploring their geometric properties and applications to dynamical systems in non-Archimedean settings.

Contribution

It provides an elementary characterization of unicritical rational functions via continued fractions and analyzes their geometric and dynamical properties.

Findings

Existence of unicritical rational functions in positive characteristic

Characterization of these functions using continued fractions

Insights into the geometry of the space of unicritical functions

Abstract

Over an algebraically closed field of positive characteristic, there exist rational functions with only one critical point. We give an elementary characterization of these functions in terms of their continued fraction expansions. Then we use this tool to discern some of the basic geometry of the space of unicritical rational functions, as well as its quotients by the SL(2)-actions of conjugation and postcomposition. We also give an application to dynamical systems with restricted ramification defined over non-Archimedean fields of positive residue characteristic.