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The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc

Steven R. Bell·Faisal Kaleem
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Abstract

Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that is an n-to-one branched covering with the properties that it extends continuously to the boundary and maps each boundary curve one-to-one onto the unit circle, and it maps each given point on the boundary to the point 1 in the unit circle. We 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. We also show how to generate all the proper holomorphic mappings to the unit disc via the rational function.

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Steven R. Bell, Faisal Kaleem. 2007-03-07. The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc. https://arxiv.org/abs/math/0703200

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