## Institute for Mathematical Sciences

## Preprint ims91-22

** Peter Jones**
* On Removable Sets for Sobolev Spaces in the Plane*

Abstract: Let $K$ be a compact subset of $\bar{\bold C} ={\bold R}^2$ and let $K^c$ denote its complement. We say $K\in HR$, $K$ is
holomorphically removable, if whenever $F:\bar{\bold C}
\to\bar{\bold C}$ is a homeomorphism and $F$ is holomorphic
off $K$, then $F$ is a M\"obius transformation. By composing
with a M\"obius transform, we may assume $F(\infty )=\infty$.
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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