Extension of isotopies in the plane

It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane \mathbb{C} always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h: X \times [0,1] \to \mathbb{C}, starting at the identi...

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Published in:Transactions of the American Mathematical Society 2019-10, Vol.372 (7), p.4889-4915
Main Authors: HOEHN, L. C., OVERSTEEGEN, L. G., TYMCHATYN, E. D.
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OVERSTEEGEN, L. G.
TYMCHATYN, E. D.
description It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane \mathbb{C} always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h: X \times [0,1] \to \mathbb{C}, starting at the identity, of a plane continuum X also always 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.
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