When do finite Blaschke products commute? (Q2758183)

In this paper, commuting pairs of finite Blaschke products are studied: if \(u\) is an automorphism of the unit disk \(\mathbb{D}\), and \(T_{u}: f \mapsto f\circ u\) is the composition operator associated to \(u\), then it is possible to describe completely which finite Blaschke products are eigenvectors of \(T_{u}\) (Proposition \(2.4\)). Then pairs of finite Blaschke products \((B,C)\) such that \(B\circ C=C\circ B\) are studied, which leads to counterexamples to several conjectures stated in [\textit{C. Cowen}, ``Commuting analytic functions'', Trans. Am. Math. Soc. 283, 685--695 (1984; Zbl 0542.30030)].