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A Sharp Entropy Condition For The Density Of Angular Derivatives

Alex Bergman

TL;DR

This work characterizes where a holomorphic self-map $f$ of the unit disc admits finite Carathéodory angular derivatives on a boundary arc in terms of an entropy condition on the complement. Under the local non-extremeness requirement $\int_{I} \log(1 - |f|) \, dm > -\infty$, the angular-derivative set $C(f) \cap I$ must be a countable union of Beurling-Carleson sets with finite entropy, and conversely any such union can be realized as $C(f)$ for some $f$ with $\int_{\mathbb{T}} \log(1 - |f|) \, dm > -\infty$. The proof combines the Aleksandrov disintegration of Aleksandrov-Clark measures with the BC-union characterization of Makarov and Nikolski, and provides a constructive method to obtain $f$ from a given $E$. These results connect boundary derivative behavior to entropy-structured thin sets and have implications for de Branges-Rovnyak spaces and boundary zero phenomena.

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 Carathéodory 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 Carathéodory 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.

A Sharp Entropy Condition For The Density Of Angular Derivatives

TL;DR

This work characterizes where a holomorphic self-map of the unit disc admits finite Carathéodory angular derivatives on a boundary arc in terms of an entropy condition on the complement. Under the local non-extremeness requirement , the angular-derivative set must be a countable union of Beurling-Carleson sets with finite entropy, and conversely any such union can be realized as for some with . The proof combines the Aleksandrov disintegration of Aleksandrov-Clark measures with the BC-union characterization of Makarov and Nikolski, and provides a constructive method to obtain from a given . These results connect boundary derivative behavior to entropy-structured thin sets and have implications for de Branges-Rovnyak spaces and boundary zero phenomena.

Abstract

Let be a holomorphic self-map of the unit disc. We show that if is integrable on a sub-arc of the unit circle, , then the set of points where the function f has finite Carathéodory angular derivative on I is a countable union of Beurling-Carleson sets of finite entropy. Conversely, given a countable union of Beurling-Carleson sets, , we construct a holomorphic self-map of the unit disc, , such that the set of points where the function has finite Carathéodory angular derivative is equal to and 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.
Paper Structure (3 sections, 5 theorems, 27 equations)

This paper contains 3 sections, 5 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Result

Key Result

Theorem 1

Let $f : \mathbb{D} \to \mathbb{D}$ be holomorphic and $I \subset \mathbb{T}$ an open arc. If Then $C(f) \cap I$ is a countable union of Beurling-Carleson sets. Conversely, if $E \subset \mathbb{T}$ is a countable union of Beurling-Carleson sets there exists an analytic function $f : \mathbb{D} \to \mathbb{D}$ satisfying and $C(f) = E$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • Theorem 3
  • Lemma 2
  • proof
  • Remark 2
  • proof : Proof of Theorem \ref{['thm:local_ang_deriv']}