The basin of infinity of tame polynomials

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.