Semi-groups of analytic maps
Given a family \(F\) of analytic self-maps of a hyperbolic domain \(D\subset\mathbb{C}\), let \(F_0\) denote the collection of all finite compositions of functions from \(F\). The following theorem is proved: If the closure of \(F_0\) (in the topology of locally uniform convergence) contains neither the identity map nor constant maps whose value lies in \(\partial D\), then any composition sequence of \(F\) converges (locally uniformly) to a constant. This is an extension of a result from [\textit{I. N. Baker} and \textit{P. J. Rippon}, Complex Variables, Theory Appl. 12, 181--200 (1989; Zbl 0644.30014)] on self-maps of the unit disc (which in turn generalizes the Denjoy-Wolff theorem).
scientific article; zbMATH DE number 1833067