Extending isotopies of planar continua

Lex G. Oversteegen¹ ; Edward D. Tymchatyn²

¹Department of Mathematics<br/>University of Alabama at Birmingham<br/>Birmingham, AL 35294<br/>United States
²Department of Mathematics and Statistics<br/>University of Saskatchewan<br/>106 Wiggins Road<br/>Saskatoon, Canada, S7N 5E6

Annals of mathematics, Tome 172 (2010) no. 3, pp. 2105-2133

Résumé

In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\mathbb{C}$ and suppose that $h:Z\times [0,1]\to \mathbb{C}$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that accessibility of a point is preserved during the isotopy. We show next that the isotopy can be extended over small hyperbolic crosscuts which are shown to remain small under the isotopy. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an isotopy. It also relies on the existence of a partition of a hyperbolic, simply connected domain $U$ in the sphere, into hyperbolically convex subsets, which have limited distortion under conformal maps to the unit disk.

Détail

DOI : 10.4007/annals.2010.172.2105