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Transactions of the American Mathematical Society

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Extension of isotopies in the plane


Authors: L. C. Hoehn, L. G. Oversteegen and E. D. Tymchatyn
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 57N37, 54C20; Secondary 57N05, 54F15
DOI: https://doi.org/10.1090/tran/7820
Published electronically: June 17, 2019
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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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Additional Information

L. C. Hoehn
Affiliation: Department of Computer Science & Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, Ontario, Canada, P1B 8L7
Email: loganh@nipissingu.ca

L. G. Oversteegen
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
Email: overstee@uab.edu

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins road, Saskatoon, Canada, S7N 5E6
Email: tymchat@math.usask.ca

DOI: https://doi.org/10.1090/tran/7820
Keywords: Isotopy, extension, plane, holomorphic motion
Received by editor(s): April 24, 2018
Received by editor(s) in revised form: December 17, 2018
Published electronically: June 17, 2019
Additional Notes: The first named author was partially supported by NSERC grant RGPIN 435518.
The second named author was partially supported by NSF-DMS-1807558.
The third named author was partially supported by NSERC grant OGP-0005616.
Article copyright: © Copyright 2019 American Mathematical Society