On the convergence of boundary points for hyperbolic inner functions

TL;DR

This paper studies how boundary points of hyperbolic inner functions converge to the Denjoy-Wolff point, providing explicit convergence rate bounds based on the angular derivative, including cases with singularities.

Contribution

It offers explicit bounds on convergence rates for boundary points of hyperbolic inner functions, extending to cases with singular Denjoy-Wolff points.

Findings

Almost every boundary point converges to the Denjoy-Wolff point under iteration.

Explicit bounds relate convergence rate to the angular derivative.

Results include cases where the Denjoy-Wolff point is a singularity.

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.