A Sharp Entropy Condition for the Density of Angular Derivatives

Alex Bergman

doi:10.5802/crmath.695

Research article - Operator theory

A Sharp Entropy Condition for the Density of Angular Derivatives

Alex Bergman ¹

¹ Matematikcentrum, Sölvegatan 18A, 22100 Lund, Sweden

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 29-33.

Abstract (Original language)
Résumé

Let $f$ be a holomorphic self-map of the unit disc. We show that if $\log (1-\vert f(z)\vert )$ 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 $f$ function has finite Carathéodory angular derivative is equal to $E$ and $\log (1-\vert f(z)\vert )$ 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.

Soit $f$ une auto-application holomorphe du disque unitaire. Nous montrons que si $\log (1-\vert f(z)\vert )$ est intégrable sur un sous-arc du cercle unitaire, $I$, alors l’ensemble des dérivées angulaires de Carathéodory de $f$ sur $I$ est une union dénombrable d’ensembles de Beurling–Carleson d’entropie finie. Inversement, étant donné une union dénombrable d’ensembles de Beurling–Carleson, $E$, nous construisons une auto-application holomorphe du disque unitaire, telle que son ensemble de dérivées angulaires de Carathéodory est égal à $E$ et que $\log (1-\vert f(z)\vert )$ est intégrable sur le cercle unitaire. Nos principaux outils techniques sont le théorème de désintégration d’Aleksandrov et une caractérisation des unions dénombrables d’ensembles de Beurling–Carleson due à Makarov et Nikolski.

Received: 2024-09-22
Revised: 2024-10-17
Accepted: 2024-10-21
Published online: 2025-03-06

DOI: 10.5802/crmath.695

Classification: 30J99, 30C35, 30H10, 46E22
Keywords: Angular derivative, holomorphic map, Beurling–Carleson sets
Mots-clés : Dérivée angulaire, carte holomorphe, ensembles de Beurling–Carleson

Author's affiliations:

Alex Bergman ¹

¹ Matematikcentrum, Sölvegatan 18A, 22100 Lund, Sweden

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {A {Sharp} {Entropy} {Condition} for the {Density} of {Angular} {Derivatives}},
     journal = {Comptes Rendus. Math\'ematique},
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     doi = {10.5802/crmath.695},
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Alex Bergman. A Sharp Entropy Condition for the Density of Angular Derivatives. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 29-33. doi : 10.5802/crmath.695. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.695/

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