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

Abstract

Let $\varphi:\mathbb D \to \mathbb D$ 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 $\varphi$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $\psi\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $\varphi$ if and only if the two self-maps have the same Denjoy-Wolff point and $\psi$ is a pseudo-iterate of $\varphi$ in the sense of Cowen. Moreover, we show that the centralizer of $\varphi$, i.e. the semigroup $\mathcal Z_\forall(\varphi):=\{\psi:\psi\circ\varphi=\varphi\circ\psi\}$ is commutative. We also prove that if $\varphi$ is univalent, then all elements of $\mathcal Z_\forall(\varphi)$ are univalent as well, and if $\varphi$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(\varphi)$. 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.