A note on forward iteration of inner functions

TL;DR

This paper investigates the behavior of compositions of inner functions fixing the origin, showing limits are either constant or inner, and explores convergence properties and counterexamples related to boundary extensions.

Contribution

It establishes that limits of compositions of such inner functions are either constant or inner, and provides conditions for boundary extension convergence.

Findings

Limit functions are either constant or inner.

Certain conditions ensure $L^1$ convergence of boundary extensions.

Counterexample shows divergence without extra 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 $\partial\mathbb{D}$.