insight - Scientific Computing - # Carathéodory Angular Derivatives

A Sharp Entropy Condition for the Existence of Carathéodory Angular Derivatives of Holomorphic Self-Maps of the Unit Disc

Q: How does the concept of entropy relate to the distribution of singularities of holomorphic functions in more general settings?

The concept of entropy, particularly as applied to Beurling-Carleson sets, provides a quantitative measure of the "thinness" or "concentration" of a set on the boundary of a domain. In the context of holomorphic functions, this translates to understanding the distribution of singularities. General Principle: Higher entropy suggests a more spread-out set, while lower entropy indicates a higher degree of concentration. For singularities of holomorphic functions, this means that sets with lower entropy correspond to functions whose singularities are more densely clustered. Beyond the Unit Disc: While the paper focuses on the unit disc, the connection between entropy and singularity distribution extends to more general settings: Simply Connected Domains: The Riemann mapping theorem allows us to relate the unit disc to other simply connected domains. Beurling-Carleson sets and their entropy can be transferred via this mapping, preserving their connection to the behavior of holomorphic functions. Multiple Variables: In several complex variables, the notion of Beurling-Carleson sets has generalizations, and entropy-like conditions appear in the study of zero sets and singular sets of holomorphic functions. Examples: Blaschke Products: The zeros of a Blaschke product (a type of holomorphic self-map of the unit disc) form a Beurling-Carleson set. The entropy of this set relates to the growth rate of the Blaschke product. Inner Functions: More generally, the singular sets of inner functions (holomorphic self-maps of the disc with boundary values of modulus 1 almost everywhere) can be characterized using entropy conditions.

Q: Could there be alternative characterizations of the sets where Carathéodory angular derivatives exist, perhaps using different geometric or analytic conditions?

Yes, alternative characterizations of sets where Carathéodory angular derivatives exist are possible, going beyond the entropy condition presented in the paper. Here are some potential avenues: Geometric Characterizations: Hausdorff Dimension: Instead of entropy, one could explore the Hausdorff dimension of the set C(f) where angular derivatives exist. There might be a critical dimension related to the existence of angular derivatives. Porosity: The notion of porosity measures the "holes" in a set. Sets with high porosity have more gaps, potentially relating to the distribution of angular derivatives. Analytic Conditions: Capacity: Capacities (like logarithmic capacity) provide alternative ways to measure the size of sets. Conditions on the capacity of C(f) might be linked to the existence of angular derivatives. Integral Conditions: Instead of log-integrability, one could investigate other integral conditions on the function (1 - |f(z)|) or related quantities to characterize C(f). Dynamical Characterizations: Julia Sets: For holomorphic self-maps, the Julia set captures the chaotic dynamics of the map. The relationship between C(f) and the Julia set of f could provide insights.

Q: If we consider the dynamics of iterating the holomorphic self-map, how does the entropy condition influence the long-term behavior of points on the unit circle?

The entropy condition, by constraining the set C(f) where angular derivatives exist, indirectly influences the long-term behavior of points on the unit circle under iteration of the holomorphic self-map f. Attracting Behavior near C(f): Points on the unit circle near C(f) tend to be attracted, under iteration of f, towards the points where angular derivatives exist. This is because the angular derivative essentially measures the "expansion" or "contraction" of the map near the boundary. Repelling Behavior Elsewhere: Points far from C(f) are less influenced by the angular derivative condition and may exhibit more complicated dynamics. Entropy and Spreading: Low Entropy: If C(f) has low entropy (highly concentrated), the iterates of most points on the unit circle will tend to cluster around a small number of attracting points. High Entropy: If C(f) has high entropy (more spread out), the dynamics of f on the unit circle might be more complicated, with points potentially spreading out more under iteration. Connections to Ergodic Theory: The study of the long-term behavior of iterates of a map is a central theme in ergodic theory. The entropy condition on C(f) could potentially be related to ergodic properties of the map f on the unit circle, such as the existence of invariant measures and their entropy.

Core Concepts

For a holomorphic self-map of the unit disc to have a finite Carathéodory angular derivative on a sub-arc of the unit circle, the set of points where this occurs must be a countable union of Beurling-Carleson sets of finite entropy.

Abstract

Bibliographic Information:

Bergman, A. (2024). A Sharp Entropy Condition For The Density Of Angular Derivatives. arXiv preprint arXiv:2409.14389v2.

Research Objective:

This paper investigates the necessary and sufficient conditions for the existence of Carathéodory angular derivatives of holomorphic self-maps of the unit disc, focusing on the entropy condition of the set where these derivatives exist.

Methodology:

The author utilizes the Aleksandrov disintegration Theorem and a characterization of countable unions of Beurling-Carleson sets by Makarov and Nikolski to prove the main theorem. The proof involves analyzing the absolute continuity of Aleksandrov-Clark measures and relating them to the existence of Carathéodory angular derivatives.

Key Findings:

The paper proves that if a holomorphic self-map of the unit disc is locally non-extreme on a sub-arc of the unit circle, the set of points where it has a finite Carathéodory angular derivative is a countable union of Beurling-Carleson sets. Conversely, for any countable union of Beurling-Carleson sets, there exists a non-extreme holomorphic self-map whose set of points with finite Carathéodory angular derivatives coincides with the given set.

Main Conclusions:

The research establishes a sharp entropy condition for the existence of Carathéodory angular derivatives of holomorphic self-maps of the unit disc. This condition implies that the derivatives cannot be arbitrarily dispersed and must exhibit a certain degree of clustering.

Significance:

This result contributes significantly to the understanding of the behavior of holomorphic functions and their boundary properties. It has implications for various areas of analysis, including perturbation theory of operators, conformal mapping, and the study of model and de Branges-Rovnyak spaces.

Limitations and Future Research:

The paper focuses on holomorphic self-maps of the unit disc. Future research could explore extending these results to more general domains or classes of functions. Additionally, investigating the implications of this entropy condition in specific applications of Carathéodory angular derivatives could be of interest.

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A Sharp Entropy Condition For The Density Of Angular Derivatives

by Alex Bergman at arxiv.org 11-06-2024

https://arxiv.org/pdf/2409.14389.pdf

A Sharp Entropy Condition For The Density Of Angular Derivatives

Deeper Inquiries

How does the concept of entropy relate to the distribution of singularities of holomorphic functions in more general settings?

The concept of entropy, particularly as applied to Beurling-Carleson sets, provides a quantitative measure of the "thinness" or "concentration" of a set on the boundary of a domain. In the context of holomorphic functions, this translates to understanding the distribution of singularities.

General Principle: Higher entropy suggests a more spread-out set, while lower entropy indicates a higher degree of concentration. For singularities of holomorphic functions, this means that sets with lower entropy correspond to functions whose singularities are more densely clustered.

Beyond the Unit Disc: While the paper focuses on the unit disc, the connection between entropy and singularity distribution extends to more general settings:

Simply Connected Domains: The Riemann mapping theorem allows us to relate the unit disc to other simply connected domains. Beurling-Carleson sets and their entropy can be transferred via this mapping, preserving their connection to the behavior of holomorphic functions.
Multiple Variables: In several complex variables, the notion of Beurling-Carleson sets has generalizations, and entropy-like conditions appear in the study of zero sets and singular sets of holomorphic functions.

Examples:

Blaschke Products:  The zeros of a Blaschke product (a type of holomorphic self-map of the unit disc) form a Beurling-Carleson set. The entropy of this set relates to the growth rate of the Blaschke product.
Inner Functions: More generally, the singular sets of inner functions (holomorphic self-maps of the disc with boundary values of modulus 1 almost everywhere) can be characterized using entropy conditions.

Could there be alternative characterizations of the sets where Carathéodory angular derivatives exist, perhaps using different geometric or analytic conditions?

Yes, alternative characterizations of sets where Carathéodory angular derivatives exist are possible, going beyond the entropy condition presented in the paper. Here are some potential avenues:

Geometric Characterizations:

Hausdorff Dimension: Instead of entropy, one could explore the Hausdorff dimension of the set C(f) where angular derivatives exist.  There might be a critical dimension related to the existence of angular derivatives.
Porosity: The notion of porosity measures the "holes" in a set. Sets with high porosity have more gaps, potentially relating to the distribution of angular derivatives.

Analytic Conditions:

Capacity:  Capacities (like logarithmic capacity) provide alternative ways to measure the size of sets.  Conditions on the capacity of C(f) might be linked to the existence of angular derivatives.
Integral Conditions:  Instead of log-integrability, one could investigate other integral conditions on the function (1 - |f(z)|) or related quantities to characterize C(f).

Dynamical Characterizations:

Julia Sets:  For holomorphic self-maps, the Julia set captures the chaotic dynamics of the map. The relationship between C(f) and the Julia set of f could provide insights.

If we consider the dynamics of iterating the holomorphic self-map, how does the entropy condition influence the long-term behavior of points on the unit circle?

The entropy condition, by constraining the set C(f) where angular derivatives exist, indirectly influences the long-term behavior of points on the unit circle under iteration of the holomorphic self-map f.

Attracting Behavior near C(f):  Points on the unit circle near C(f) tend to be attracted, under iteration of f, towards the points where angular derivatives exist. This is because the angular derivative essentially measures the "expansion" or "contraction" of the map near the boundary.

Repelling Behavior Elsewhere: Points far from C(f) are less influenced by the angular derivative condition and may exhibit more complicated dynamics.

Entropy and Spreading:

Low Entropy: If C(f) has low entropy (highly concentrated), the iterates of most points on the unit circle will tend to cluster around a small number of attracting points.
High Entropy: If C(f) has high entropy (more spread out), the dynamics of f on the unit circle might be more complicated, with points potentially spreading out more under iteration.

Connections to Ergodic Theory:  The study of the long-term behavior of iterates of a map is a central theme in ergodic theory. The entropy condition on C(f) could potentially be related to ergodic properties of the map f on the unit circle, such as the existence of invariant measures and their entropy.