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
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