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 b _j 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 b _j 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 f ₁ and f ₂ 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.