Extension of isotopies in the plane

Hoehn, L.; Oversteegen, L.; Tymchatyn, E.

doi:10.1090/tran/7820

Extension of isotopies in the plane
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by L. C. Hoehn, L. G. Oversteegen and E. D. Tymchatyn PDF

Trans. Amer. Math. Soc. 372 (2019), 4889-4915 Request permission

Abstract:

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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