Semi-groups of analytic maps
type: Item
instance of: scholarly article
title: Semi-groups of analytic maps
zbMATH Open document ID: 1015.30010
DOI: 10.1007/BF03320988
author: Alan F. Beardon
published in: Computational Methods and Function Theory
publication date: 2001-01-01T00:00:00Z
review text: Given a family \(F\) of analytic self-maps of a hyperbolic domain \(D\subset\mathbb{C}\), let \(F_0\) denote the collection of all finite compositions of functions from \(F\). The following theorem is proved: If the closure of \(F_0\) (in the topology of locally uniform convergence) contains neither the identity map nor constant maps whose value lies in \(\partial D\), then any composition sequence of \(F\) converges (locally uniformly) to a constant. This is an extension of a result from [\textit{I. N. Baker} and \textit{P. J. Rippon}, Complex Variables, Theory Appl. 12, 181--200 (1989; Zbl 0644.30014)] on self-maps of the unit disc (which in turn generalizes the Denjoy-Wolff theorem).
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30C80
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 1833067
zbMATH Keywords: composition of analytic maps
zbMATH Keywords: hyperbolic domain
zbMATH Keywords: Denjoy-Wolff theorem
MaRDI profile type: Publication
cites work: Q4177796
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full work available at URL: https://doi.org/10.1007/bf03320988
OpenAlex ID: W2058867828
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description: scientific article; zbMATH DE number 1833067