A note on forward iteration of inner functions
Gustavo Rodrigues Ferreira
TL;DR
This work addresses forward iteration of inner holomorphic self-maps of the unit disk and establishes a dichotomy for locally uniform limits: any limit $F$ of the forward iterates $F_n=f_n\circ\cdots\circ f_1$ is either constant or inner, with non-constant limits characterized by $\sum_{n\ge1} (1 - |f_n'(0)|) < \infty$. In the Blaschke special case, the same dichotomy holds and, under a Frostman-type summability condition $\sum_{n\ge1} (1 - b_n'(0)) \log\frac{1}{1 - b_n'(0)} < \infty$ (with uniformly bounded degree growth), the boundary extensions converge in $L^1$ to the boundary limit; without this condition, boundary values may diverge on the entire boundary. The paper also proves that zeros of the iterates satisfy $Z(B_n)\subset Z(B_{n+1})$ and $Z(B)=\cup_n Z(B_n)$, implying the Blaschke condition $\sum_{z\in Z(B)} (1 - |z|) < \infty$ for the limit Blaschke product. A constructive counterexample shows Frostman failure leads to boundary divergence, highlighting the sharpness of the convergence criteria. Overall, the results connect forward-iteration dynamics of inner functions to zero-set structure and boundary behavior through precise summability and Frostman-type conditions.
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 $f_n:\mathbb{D}\to\mathbb{D}$ are inner functions fixing the origin, we show that a limit function of $f_n\circ\cdots\circ f_1$ 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 $L^1$ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of $\partial\mathbb{D}$.