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arXiv · 2207.05198

Continuous analytic capacity and holomorphic motions

Malik Younsi
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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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Malik Younsi. 2022-07-11. Continuous analytic capacity and holomorphic motions. https://arxiv.org/abs/2207.05198

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