On the convergence of boundary points for hyperbolic inner functions

Abstract

Given a hyperbolic inner function $f \colon \mathbb{D} \to \mathbb{D}$ with Denjoy-Wolff point $p \in \partial \mathbb{D}$, it is well known that almost every point $\xi\in \partial \mathbb{D}$ converges to $p$ under iteration of the radial extension $f^* \colon \partial \mathbb{D} \to \partial \mathbb{D}$. We provide explicit bounds for the rate of this convergence in terms of the angular derivative, holding almost surely. Our results also cover the case where the Denjoy-Wolff point is a singularity.