The basin of infinity of tame polynomials

TL;DR

This paper characterizes when two tame polynomials over non-Archimedean fields are analytically conjugate on their basin of infinity, and describes the closure of the tame shift locus in terms of Fatou sets.

Contribution

It provides a necessary and sufficient condition for conjugacy of tame polynomials and characterizes the closure of the tame shift locus via Fatou sets.

Findings

Two tame polynomials are conjugate iff their dynamics on the basin of infinity are analytically equivalent.

The tame shift locus consists of polynomials with all critical points in the basin of infinity.

The closure of the tame shift locus includes polynomials whose Fatou set equals their basin of infinity.

Abstract

Let $\mathbb{C}_v$ be a characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in $\mathbb{C}_v[z]$ of degree $d \ge 2$ to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials, tame polynomials with all their critical points in the basin of infinity form the tame shift locus. We show that a tame map $f\in\mathbb{C}_v[z]$ is in the closure of the tame shift locus if and only if the Fatou set of $f$ coincides with the basin of infinity.