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

TL;DR

This paper characterizes the holomorphic self-maps commuting with a zero hyperbolic step parabolic map of the unit disc, showing they share the same Denjoy-Wolff point and are pseudo-iterates, with a focus on their algebraic and geometric properties.

Contribution

It provides a complete description of the centralizer of such maps, proving commutativity and univalence preservation, and introduces a new approach using simultaneous linearization techniques.

Findings

Commuting maps share the same Denjoy-Wolff point.

The centralizer forms a commutative semigroup.

Univalence of maps is preserved in the centralizer.

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.