The Structure of the Semigroup of Proper Holomorphic Mappings of a Planar Domain to the Unit Disc

Abstract

Given a bounded n-connected domain Ω in the plane bounded by n non-intersecting Jordan curves and given one point bj on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping f of Ω onto the unit disc that is an n-to-one branched covering with the properties: f extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and f maps each given point bj on the boundary to the point 1 in the unit circle. We shall modify a proof by H. Grunsky of Bieberbach’s result to show that there is a rational function of 2n + 2 complex variables that generates all of these maps. In fact, we show that there are two Ahlfors maps f1 and f2 associated with the domain such that any such mapping is given by a fixed linear fractional transformation mapping the right half plane to the unit disc composed with c R + i C, where R is a rational function of the 2n + 2 functions \(f_1(z),f_2(z), and f_1(b_1), f_2(b_1),...f_1(b_n),f_2(b_n)\), and where c and C are arbitrary real constants subject to the condition c > 0. We also show how to generate all the proper holomorphic mappings to the unit disc via the rational function R.