Random iteration of holomorphic self-maps over bounded domains in \(C^ N\)
type: Item
instance of: scholarly article
title: Random iteration of holomorphic self-maps over bounded domains in \(C^ N\)
zbMATH Open document ID: 0826.32021
author: Fuyao Ren
author: Wenjun Zhang
published in: Chinese Annals of Mathematics. Series B
publication date: 1995-11-19T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{C}^N\), and let \(\{f_n\}\) be a sequence of holomorphic mappings from \(\Omega\) onto itself. The authors give sufficient conditions for the compositions \(F_n = f_n \circ f_{n-1} \circ \cdots \circ f_1\) (respectively \(G_n = f_1 \circ f_2 \circ \cdots \circ f_n)\) to converge to a constant mapping. For example, if \(f_n(\Omega) \subset E \subset \subset \Omega\) for all \(n\), then \(\{G_n\}\) converges to a point of \(\Omega\), but \(\{F_n\}\) may not. On the other hand, if \(\Omega\) is taut and \(\{f_n\}\) converges to a mapping having an attracting fixed point in \(\Omega\), then both \(\{F_n\}\) and \(\{G_n\}\) converge to points of \(\Omega\). The authors' proofs, which employ the Kobayashi distance, seem simpler than those given previously for simply connected domains in \(\mathbb{C}\).
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 32F45
zbMATH DE Number: 743969
zbMATH Keywords: random iteration
zbMATH Keywords: Kobayashi distance
reviewed by: Theodore J. Barth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 743969