Extension of isotopies in the plane

TL;DR

This paper characterizes when isotopies of certain plane compact sets can be extended to the entire plane, showing that uniform bounds on component diameters guarantee such extensions.

Contribution

It provides a precise characterization for extending isotopies of uniformly perfect plane compacta and proves the extension is always possible under uniform component size conditions.

Findings

Extension is always possible if component diameters are bounded away from zero.

Characterization applies specifically to uniformly perfect plane compacta.

Not all plane compacta allow isotopy extension, only those meeting the criteria.

Abstract

Let $A$ be any plane set. It is known that a holomorphic motion $h: A \times \mathbb{D} \to \mathbb{C}$ always extends to a holomorphic motion of the entire plane. It was recently shown that any isotopy $h: X \times [0,1] \to \mathbb{C}$, starting at the identity, of a plane continuum $X$ also 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.