Extending Isotopies of Planar Continua

TL;DR

This paper proves that isotopies of planar continua can be extended to the entire plane, introducing a new characterization of accessible points and utilizing metric external rays for the extension process.

Contribution

It provides a novel method for extending isotopies of planar continua to the whole plane using hyperbolic crosscuts and metric external rays.

Findings

Accessible points are preserved during isotopies.

Isotopies can be extended over hyperbolic crosscuts.

The approach simplifies controlling isotopies via metric external rays.

Abstract

In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum 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 of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that an accessible point is preserved during the isotopy. We show next that the isotopy can be extended over hyperbolic crosscuts. 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.