Continuous analytic capacity and holomorphic motions

2022/07/11 by Malik Younsi, Younsi, Malik
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2207.05198

openalex publication_date 2022/07/11 · openalex created_date 2022/07/15 · openalex updated_date 2026/07/28

Abstract

We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop--Carleson--Garnett--Jones and Browder--Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist.

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