Commuting inner functions
Let \(\mathbb{D}\) denote the open unit disk in \(\mathbb{C}\) and let \(S_n\) denote the \(n\)th iterate of the atomic inner function \(S\) given by \(S(z)= \exp(-\frac{1+z} {1-z})\) on \(\mathbb{D}\). The author determines all analytic selfmappings of the disk \(\mathbb{D}\) commuting with \(S_n\). It turns out that these are only the obvious ones: the identity, a certain constant mapping or an iterate \(S_m\) for an integer \(m\). The proof makes clever use of the Denjoy-Wolff theorem and the Schwarz-Pick lemma. Recall that for any inner function \(u\) the Frostman shift \(\frac{a-u} {1-\overline{a}u}\) is in fact a Blaschke product for all \(a\in\mathbb{D}\) outside a set of logarithmic capacity zero. But when is this Blaschke product an interpolating one? In the second part of the paper the author proves that the Frostman shift of \(S_n\) is an interpolating Blaschke product if and only if \(a\notin \{S_j(0): j=1,2,\dots, n- 1\}\cup\{0\}\).
scientific article; zbMATH DE number 1031631