Hypercyclicity of composition operators in Stein manifolds (Q2814417)

This interesting paper investigates hypercyclic properties of composition operators \(C_{\varphi}: f \mapsto f \circ \varphi\) on the space \(H(\Omega)\) of holomorphic functions on a connected Stein manifold \(\Omega\), for a holomorphic self-map \(\varphi: \Omega \rightarrow \Omega\). When \(\Omega\) is a domain in the complex plane, a rather complete characterization of the hypercyclicity of \(C_{\varphi}\) was presented by \textit{K.-G. Grosse-Erdmann} and \textit{R. Mortini} [J. Anal. Math. 107, 355--376 (2009; Zbl 1435.30159)]. Exploiting ideas of Grosse-Erdmann and Mortini [loc. cit.] and \textit{L. Bernal-González} [J. Math. Anal. Appl. 305, No. 2, 690--697 (2005; Zbl 1071.32002)] and utilizing the Oka-Weil theorem, equivalent conditions for hypercyclicity and hereditary hypercyclicity of \(C_{\varphi}\) with respect to a given sequence of natural numbers are presented. In some class of higher-dimensional Stein manifolds, the conditions obtained by the author are simplified, and they take the same form as in planar domains. This family of manifolds consists of those connected Stein manifolds \(\Omega\) in which all balls with respect to the Carathéodory pseudo-distance are relatively compact in the topology of \(\Omega\). There are many examples of manifolds in this class. Moreover, it is shown that every hypercyclic operator \(C_{\varphi}\) on \(H(\Omega)\), with \(\Omega\) in this class, is hereditarily hypercyclic.