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arXiv · 2206.00374

A note on forward iteration of inner functions

Gustavo Rodrigues Ferreira
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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 $\partial\mathbb{D}$.

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Gustavo Rodrigues Ferreira. 2022-06-01. A note on forward iteration of inner functions. https://doi.org/10.1112/blms.12779

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