Extension of isotopies in the plane

Academic Article

Abstract

  • © 2019 American Mathematical Society It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane C always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h : X×[0, 1] → 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.
  • Authors

    Digital Object Identifier (doi)

    Author List

  • Hoehn LC; Oversteegen LG; Tymchatyn ED
  • Start Page

  • 4889
  • End Page

  • 4915
  • Volume

  • 372
  • Issue

  • 7