Contemporary Mathematics

Volume 355, 2004

Barycentric extensions of monotone maps of the circle

William Abikoff, Clifford J. Earle, and Sudeb Mitra

ABSTRACT. There is a natural approach to barycentric extension based on

the MAY iterator. Precisely, we define the conformally natural extension

by defining a class of conformally natural dynamical systems acting anti-

holomorphically on the unit disk.

The natural class of functions admitting extension are the continuous

monotone degree ±1 functions on the circle. Those functions are cell-collapse

maps: they contract some disjoint closed intervals to points but are otherwise

homeomorphisms.

Here we show that the barycentric extension of a cell-collapse map of

the circle is itself a cell-collapse map in the closed disk. On the interior of

the hyperbolic convex hull of the complement of the collapsed intervals, the

extension is a real analytic diffeomorphism onto the open unit disk. This

theorem is proved by showing the validity of the MAY algorithm for computing

the extension.

An example is given in which the boundary function is based on a con-

struction of Lebesgue. The boundary function is locally constant on the com-

plement of a Cantor set.

In an appendix, we derive a needed generalization of the Denjoy-Wolff

Theorem to maps which are either isometries or contractive in the hyperbolic

metric.

1.

Introduction

Let~

be the open unit disk, T the unit circle,

and~

the closed unit disk

~UT.

There is a long history to the relationship between homeomorphisms

ofT

and their

extensions to homeomorphisms

of~.

Early results include the Alexander extension

and the Beurling-Ahlfors extension transported from the upper half-plane; the latter

is quasiconformal when possible.

In the opposite direction, Nielsen analyzed surface diffeomorphisms by repre-

senting the surface as the quotient of

~

by a discrete group of conformal automor-

phisms, lifting the diffeomorphisms to

~.

and studying the boundary values of the

lifts. In the cases Nielsen considered, these boundary values are homeomorphisms

of T that respect the conformal group action.

2000 Mathematics Subject Classification. Primary 30C30, 37E10; Secondary 54C20, 65E05,

30C15.

The research .of the third author was partially supported by the CUNY Research Foundation

under grant PSC-CUNY 65451-0034.

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/355/06442