# Extending Isotopies of Planar Continua - Mathematics > Geometric Topology

Abstract: In this paper we solve the following problem in the affirmative: Let $Z$ be acontinuum in the plane $\complex$ and suppose that $h:Z\times 0,1\to\complex$is an isotopy starting at the identity. Can $h$ be extended to an isotopy ofthe plane?We will provide a new characterization of an accessible point in a planarcontinuum $Z$ and use it to show that an accessible point is preserved duringthe isotopy. We show next that the isotopy can be extended over hyperboliccrosscuts. The proof makes use of the notion of a metric external ray, whichmimics the notion of a conformal external ray, but is easier to control duringan isotopy.

Author: Lex G. Oversteegen, E.D. Tymchatyn

Source: https://arxiv.org/