A note on forward iteration of inner functions

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 $\mathrm{\partial }\mathbb{D}$.