On holomorphic maps which commute with hyperbolic automorphisms
type: Item
instance of: scholarly article
title: On holomorphic maps which commute with hyperbolic automorphisms
zbMATH Open document ID: 0976.32014
author: Chiara de Fabritiis
author: Graziano Gentili
published in: Advances in Mathematics
publication date: 1999-11-11T00:00:00Z
review text: Suppose \(\gamma\) is a holomorphic automorphism of the unit disk \(\Delta\) in \({\mathbb C}\) and suppose \(f\in \text{Hol}(\Delta,\Delta)\) with \(f \circ \gamma = \gamma \circ f\). \textit{M. H. Heins} [Duke Math. J. 8, 312-316 (1941; Zbl 0025.17301)] showed that either \(f = \text{id}\) or \(f\) is also a hyperbolic automorphism of \(\Delta\) with the same fixed points as \(\gamma\). This paper provides some insight into a similar problem for \(\Delta_n\), the unit ball in \({\mathbb C}^n\), when \(n>1\). The first observation is that the two fixed points \(p_1\) and \(p_2\) of \(\gamma\) are ``fixed points'' of \(f\), in the sense of \(K\)-limits. By conjugating in \(\text{Aut}(\Delta_n)\) we may suppose that these fixed points are \(e_1\) and \(-e_1\). Under the assumption of a ``regularity condition'', i.e., the existence of a certain \(K\)-limit at \(e_1\), the authors show that \(f_1\) (the first component function of \(f\)) does not depend on \(z_2, \ldots, z_n\). An explicit formula for \(f_1\) is given.
reviewed by: Bruce Gilligan
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 1334585
zbMATH Keywords: hyperbolic automorphisms
zbMATH Keywords: Julia-Wolff-Carathéodory Theorem
MaRDI profile type: Publication
full work available at URL: aima.1998.1806
OpenAlex ID: W2020433983
cites work: Q3999640
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DOI: 10.1006/AIMA.1998.1806
description: scientific article; zbMATH DE number 1334585