Wandering Fatou Components and Algebraic Julia Sets

TL;DR

This paper investigates non-Archimedean polynomial dynamics, showing wandering Fatou components are contained in basins of periodic orbits, and characterizes when Julia sets are algebraic based on critical point recurrence.

Contribution

It provides a complete description of Julia set points in the Berkovich line and characterizes algebraic Julia sets via critical point behavior.

Findings

Wandering Fatou components lie within basins of periodic orbits.

New Julia set points emerge when passing to the Berkovich line.

Polynomials with algebraic Julia sets have nonrecurrent critical points.

Abstract

We study the dynamics of polynomials with coefficients in a non-Archimedean field $K,$ where $K$ is a field containing a dense subset of algebraic elements over a discrete valued field $k.$ We prove that every wandering Fatou component is contained in the basin of a periodic orbit. We obtain a complete description of the new Julia set points that appear when passing from $K$ to the Berkovich line over $K$. We give a dynamical characterization of polynomials having algebraic Julia sets. More precisely, we establish that a polynomial with algebraic coefficients has algebraic Julia set if every critical element is nonrecurrent.