INFINITE BRANCHED CIRCLE PACKINGS AND DISCRETE COMPLEX POLYNOMIALS

Abstract

In the last decade much research has been done on questions related to circle packings, which are collections of circles in the plane with prescribed patterns of tangencies encoded by simplicial 2-complexes. The interest in circle packings and their connections with analytic functions was initiated by W. Thurston in 1985 when he suggested a method for approximation of conformal mappings using circle packings [20]. This method was shown to work by B. Rodin & D. Sullivan [15] later that year (see also [17]). Since then several results have been published (for example, [3, 4, 5, 10, 11, 14]) which seem to indicate the possibility of creating a discrete analog of the theory of analytic functions via circle packings. However, most of these results deal with univalent circle packings, that is, genuine packings of circles. Influenced by the idea of [3] that circle packings might be used to construct discrete parallels of analytic functions, we started in [7] to investigate non-univalent circle packings, particularly branched packings. We introduced there the notion of discrete Blaschke products and proved a branched version of the finite Riemann mapping theorem (see [20, 15]). We feel though that to have a solid foundation for a circle packing analog of the theory of analytic functions one needs something more. Since complex polynomials form a fundamental and very important class of analytic functions, it is natural to try to establish circle packing counterparts of these mappings. This is exactly the main theme of our paper.