A note on forward iteration of inner functions

Ferreira, Gustavo R. (2023). A note on forward iteration of inner functions. Bulletin of the London Mathematical Society, 55(3) pp. 1143–1153.

DOI: https://doi.org/10.1112/blms.12779

Abstract

A well‐known problem in holomorphic dynamics is to obtain Denjoy–Wolff‐type results for compositions of self‐maps of the unit disc. Here, we tackle the particular case of inner functions: if f n : D → D $f_n:\mathbb {D}\rightarrow \mathbb {D}$ are inner functions fixing the origin, we show that a limit function of f n ∘ ⋯ ∘ f 1 $f_n\,\circ \cdots \circ\, f_1$ is either constant or an inner function. For the special case of Blaschke products, we prove a similar result and show, furthermore, that imposing certain conditions on the speed of convergence guarantees L 1 $L^1$ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of ∂ D $\partial \mathbb {D}$ .