Simultaneous models for commuting holomorphic self-maps of the ball

TL;DR

This paper establishes conditions under which multiple commuting holomorphic self-maps of the unit ball can be simultaneously conjugated to automorphisms, revealing structural insights and applications to their commutativity.

Contribution

It proves the existence of a universal conjugacy for commuting families of holomorphic self-maps of the ball, extending understanding of their structure and interactions.

Findings

Existence of simultaneous conjugacy to automorphisms

Characterization of commuting hyperbolic and parabolic maps

Universal property of the conjugacy

Abstract

We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.