Determination sets for holomorphic automorphisms of a bounded domain
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.
scientific article; zbMATH DE number 1981343