Extension of isotopies in the plane

It is known that a holomorphic motion (an analytic version of an isotopy) of a set XX in the complex plane C\mathbb {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]→Ch: X \times [0,1] \to \mathbb {C}, starting at the identity, of a plane continuum XX 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 XX 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 XX are uniformly bounded away from zero.