Random iteration of holomorphic self-maps over bounded domains in \(C^ N\) (Q1842113)

Let \(\Omega\) be a bounded domain in \(\mathbb{C}^N\), and let \(\{f_n\}\) be a sequence of holomorphic mappings from \(\Omega\) onto itself. The authors give sufficient conditions for the compositions \(F_n = f_n \circ f_{n-1} \circ \cdots \circ f_1\) (respectively \(G_n = f_1 \circ f_2 \circ \cdots \circ f_n)\) to converge to a constant mapping. For example, if \(f_n(\Omega) \subset E \subset \subset \Omega\) for all \(n\), then \(\{G_n\}\) converges to a point of \(\Omega\), but \(\{F_n\}\) may not. On the other hand, if \(\Omega\) is taut and \(\{f_n\}\) converges to a mapping having an attracting fixed point in \(\Omega\), then both \(\{F_n\}\) and \(\{G_n\}\) converge to points of \(\Omega\). The authors' proofs, which employ the Kobayashi distance, seem simpler than those given previously for simply connected domains in \(\mathbb{C}\).