On removable sets for Sobolev spaces in the plane

1991/11/26 by Peter Jones, Jones, Peter · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.math/9201298

openalex publication_date 1991/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a compact subset of \bold C =\bold R² and let K^c denote its complement. We say K∈ HR, K is holomorphically removable, if whenever F:\bold C →\bold C is a homeomorphism and F is holomorphic off K, then F is a Möbius transformation. By composing with a Möbius transform, we may assume F(∞ )=∞. The contribution of this paper is to show that a large class of sets are HR. Our motivation for these results is that these sets occur naturally (e.g. as certain Julia sets) in dynamical systems, and the property of being HR plays an important role in the Douady-Hubbard description of their structure.

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