On the convergence of boundary points for hyperbolic inner functions
Anna Jové, Mateo Mencía
TL;DR
The paper investigates the boundary dynamics of hyperbolic inner functions on the unit disk, establishing an explicit almost-sure rate of convergence of boundary points to the Denjoy–Wolff point under the radial boundary extension. The authors harness shrinking-target theory on the circle and a Möbius conjugation to convert the autonomous system into a non-autonomous one fixing 0, enabling precise probabilistic hitting results. The main result shows that for any 0<ε<1 and α=f'(p)∈(0,1), almost every boundary point eventually lies in the annulus $A(p; α^{(1+ε)n}, α^{(1-ε)n})$, providing an exponential rate of boundary convergence even when p is a singularity. The approach hinges on rate bounds for $f^n(0)$, two key shrinking-target lemmas, and the transfer of the problem to a non-autonomous fixed-point framework, with potential implications for boundary dynamics of inner functions more broadly.
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 $ξ\in \partial \mathbb{D}$ 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.