Extending isotopies of planar continua

Abstract

In this paper we solve the following problem in the affirmative: Let Z be a continuum in the plane ‫ރ‬ and suppose that h W Z OE0; 1 ! ‫ރ‬ 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.

Quasicircles and the conformal group

“…See also [15] and [8] for other applications of this construction. For the convenience of the reader we include a short and elementary proof.…”

A fixed point theorem for branched covering maps of the plane

“…an accessible point at which the ray R θ lands. Important facts concerning f -geodesics were established in [OT08]. We need to study g-geodesics and f -geodesics.…”

“…To begin with, we need a lemma which will allow us to simplify the applications of results of [FMOT07,OT08,KP94] in this section. For simplicity when talking of angles we often mean the points of S 1 with arguments equal to these angles; here S 1 is considered as the boundary of the unit disk in D ∞ .…”

“…We need to study g-geodesics and f -geodesics. As a tool we use the socalled maximal ball foliation constructed and studied in [FMOT07,OT08,KP94]. Given a crosscut T , denote the set Sh(T ) ∪ T by Sh + (T ).…”

“…In particular, a boundary g-geodesic of A maps to the corresponding boundary g-geodesic of Z A so that the endpoints are mapped as the map g prescribes. We can then extend this map over all gaps by mapping the barycenter of each gap to the barycenter of the image gap and subsequently "coning" the map on the gap (see [OT08]). …”