Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step

Abstract

Let $\ddot{I}†:\mathbb{D}\to \mathbb{D}$ be a parabolic self-map of the unit disc $\mathbb{D}$ having zero hyperbolic step. We study holomorphic self-maps of $\mathbb{D}$ commuting with $\ddot{I}†$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $\ddot{I}ˆ\in \mathsf{Hol}(\mathbb{D},\mathbb{D})$ commutes with $\ddot{I}†$ if and only if the two self-maps have the same Denjoy-Wolff point and $\ddot{I}ˆ$ is a pseudo-iterate of $\ddot{I}†$ in the sense of Cowen. Moreover, we show that the centralizer of $\ddot{I}†$, i.e. the semigroup ${\mathcal{Z}}_{\forall }(\ddot{I}†):=\{\ddot{I}ˆ:\ddot{I}ˆ\circ \ddot{I}†=\ddot{I}†\circ \ddot{I}ˆ\}$ is commutative. We also prove that if $\ddot{I}†$ is univalent, then all elements of ${\mathcal{Z}}_{\forall }(\ddot{I}†)$ are univalent as well, and if $\ddot{I}†$ is not univalent, then the identity map is an isolated point of ${\mathcal{Z}}_{\forall }(\ddot{I}†)$. The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.