Stability of Blaschke products under forward iteration

Forward iteration of holomorphic self-maps generalizes the iteration of a single function in a natural way. This framework arises in complex dynamics, for instance in the study of wandering domains and in seeking suitable extensions of the Denjoy-Wolff theorem. Here, we consider forward iteration of Blaschke products. We prove that the classes of indestructible and maximal Blaschke products are stable under forward iteration.