Continuous analytic capacity and holomorphic motions

TL;DR

The paper addresses whether continuous analytic capacity $\alpha$ varies continuously under holomorphic motions, a question inspired by known irregularities for analytic capacity $\gamma$. It introduces a construction based on holomorphic dynamics via the Böttcher motion and leverages harmonic measure and Dirichlet algebra theory to produce a counterexample showing discontinuities of $\gamma(E_\lambda)$ and $\alpha(F_\lambda)$ at $\lambda=0$. This yields a compact set whose continuous and analytic capacities fail to vary continuously under a holomorphic motion, and it also shows nonexistence of extremal functions for $\alpha$ on certain sets. Additionally, the results provide a new proof of prior capacity variation phenomena under holomorphic motions and culminate in a combined counterexample that simultaneously disrupts both $\gamma$ and $\alpha$ along the motion, highlighting intricate interactions between complex dynamics and capacity theory.

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.

Figures (1)

• Figure 1: The Böttcher motion $h_\lambda(z)$.

Theorems & Definitions (18)

• Theorem 1.1: Ransford--Younsi--Ai RYA
• Theorem 1.3
• Corollary 1.4
• Proposition 1.5
• Corollary 1.6
• Definition 2.1
• Theorem 2.2: Browder--Wermer BW
• Theorem 2.3: Bishop--Carleson--Garnett--Jones BCGJ
• Definition 2.4
• Definition 2.5