Barycentric extensions of monotone maps of the circle

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 construction of Lebesgue. The boundary function is locally constant on the complement of a Cantor set. In an appendix, we derive a needed generalization of the Denjoy-Wolff Theorem to maps which are either isometrics or contractive in the hyperbolic metric.