Determination sets for holomorphic automorphisms of a bounded domain
type: Item
instance of: scholarly article
title: Determination sets for holomorphic automorphisms of a bounded domain
zbMATH Open document ID: 1046.32001
DOI: 10.1016/S1631-073X(03)00137-7
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-15T00:00:00Z
review text: Let \(D\) be a bounded domain in \(\mathbb C\) and \(\Aut(D)\) denotes the group of analytic automorphisms of \(D\). A finite set \(W \subset D\) is called a determination set if \(f(z)=z\) for all \(z\in W\) implies \(f\equiv id\). For all \(d\in \mathbb N\), let \(W(d)\subset D^d\) be the set \((z_1, \hdots, z_d)\in D^d\) such that \((z_1, \dots , z_d)\) is a determination set for \(\Aut(D)\). The author shows that \(W(d)\) is open. \(W(d)\) could be an empty set. If \(d\geq n+1\) then it is proved that \(W(d)\) is an open set of \(D^d\) (Thm 3.2). Finally the author remarks the following topological property of determination sets. The map \(\varphi: \Aut(D) \to D^d\) given by \(f \to( f(z_1),\dots, f( z_d))\) is an homeomorphism of \(\Aut(D)\) on its image.
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32H99
zbMATH DE Number: 1981343
zbMATH Keywords: determination sets
zbMATH Keywords: holomorphic automorphisms
zbMATH Keywords: bounded domains
author: Jean-Pierre Vigué
reviewed by: Gabriela Paola Ovando
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1981343