Determination sets for analytic automorphisms and endomorphisms of a bounded domain.
The author studies the determining sets for the group \(\Aut(D)\) of the analytic automorphisms of a bounded domain \(D\subset\mathbb C^n\) (resp. for the set \(H(D,D)\) of holomorphic functions from \(D\) in \(D\)). A set of points \((z_1,\dots, z_d)\in D^d\) is called a determining set for \(\Aut(D)\) (resp. \(H(D,D)\)) if, for all \(f\in \Aut(D)\) (resp. \(H(D,D)\)), \(f(z_i) = z_i\), \(i = 1,\dots ,d\), implies \(f =\) id. The author proves that the ensemble \(W(d)\) of determining sets for \(\Aut(D)\) is an open (possibly empty) set in \(D^d\). For \(a\in D\) and \(d\geqslant n + 1\), the ensemble \(S_a(d)\) of points \((z_1,\dots , z_{d-1})\in D^{d-1}\) such that \((a,z_1,\dots ,z_{d -1})\) doesn't belong to \(W(d)\) is contained in a strict analytic subset of \(D^{d-1}\). The ensemble of determining sets for \(H(D,D)\) is an open set in \(D^d\) also. There exists an open \(U\subset D^n\) such that \((a,\dots ,a)\in \overline U\) and for all \((z_1,\dots, z_n)\in U\), \((a, z_1,\dots, z_n)\) is a determining set for \(H(D,D)\). Let \(D\) be a bounded convex domain of \(C^2\) and \((z_1,z_2)\) two distinct points of \(D\). Then there exists a nonempty open \(U\subset D\) such that for all \(z\in U\), \((z,z_1,z_2)\) is a determining set for \(H(D,D)\). Examples of determining sets for \(H(D,D)\), when \(D\) is the unity ball \(B_2\subset\mathbb C^2\) or the bidisk \(\Delta^2\subset\mathbb C^2\), are considered.
analytic automorphisms and endomorphisms
scientific article; zbMATH DE number 2162467
scientific article