A Sharp Entropy Condition For The Density Of Angular Derivatives

TL;DR

This paper characterizes the sets where holomorphic self-maps of the unit disc have finite Carathéodory angular derivatives, linking entropy conditions with geometric set properties using advanced harmonic analysis tools.

Contribution

It establishes a sharp entropy condition that precisely characterizes the sets of finite angular derivatives for such maps, and constructs examples for given sets.

Findings

Sets with finite angular derivatives are countable unions of Beurling-Carleson sets with finite entropy.

Constructs holomorphic self-maps with prescribed sets of finite angular derivatives.

Provides a characterization linking entropy conditions to geometric properties of sets.

Abstract

Let $f$ be a holomorphic self-map of the unit disc. We show that if $\log (1-\lvert f(z) \rvert)$ is integrable on a sub-arc of the unit circle, $I$, then the set of points where the function f has finite Carath\'eodory angular derivative on I is a countable union of Beurling-Carleson sets of finite entropy. Conversely, given a countable union of Beurling-Carleson sets, $E$, we construct a holomorphic self-map of the unit disc, $f$, such that the set of points where the function has finite Carath\'eodory angular derivative is equal to $E$ and $\log(1-\lvert f(z) \rvert)$ is integrable on the unit circle. Our main technical tools are the Aleksandrov disintegration Theorem and a characterization of countable unions of Beurling-Carleson sets due to Makarov and Nikolski.