Extension of isotopies in the plane

Let \(A\) be any plane set. It is known that a holomorphic motion \(h: A \times \mathbb{D} \to \mathbb{C}\) always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy \(h: X \times [0,1] \to \mathbb{C}\), starting at the identity, of a plane continuum \(X\) al...

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Bibliographic Details
Published in:arXiv.org 2018-08
Main Authors: Hoehn, L C, Oversteegen, L G, Tymchatyn, E D
Format: Article
Language:English
Subjects:
Mathematical analysis
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Summary:Let \(A\) be any plane set. It is known that a holomorphic motion \(h: A \times \mathbb{D} \to \mathbb{C}\) always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy \(h: X \times [0,1] \to \mathbb{C}\), starting at the identity, of a plane continuum \(X\) also extends to an isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta \(X\) which extend to an isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of \(X\) are uniformly bounded away from zero.
ISSN:2331-8422