Approximation Of Quasiconformal Mappings And Analytic Functions By Circle Packings

A circle packing is a configuration of circles in a constant curvature surfacewith prescribed pattern of tangencies. The progress in the field of circle pack-ings and their connections with analytic functions was initiated by W.Thurston in1985 when he suggested a method for approximation of the Riemann mapping byhexagonal circle packings. Shortly, Rodin and Sullivan proved the convergenceof Thurston's scheme, which gives a new discrete geometry view of the Riemannmapping. Since then much results on circle packings and their applications fol-lowed. For the study of circle configurations, classical circle packings consistingof disjoint open disks were generalized to circle patterns, where the disks mayoverlap. In this thesis, our main work is as follows. First, we use the methodof circle packings to investigate the approximation of quasiconformal mappings.Using the techniques of circle packings of bounded degree, we construct approx-imating solutions of quasiconformal mapping from a simply connected region tothe unit disk. It is proved that these approximating solutions on compact subsetsconverge to the exact solutions. Next, we apply the approach of branched circlepatterns to study the approximation of analytic functions with finite critical points.Given an analytic function F with finite critical points from a bounded simply-connected region onto another one, its approximating solutions are constructedby the techniques of branched circle patterns. After a suitable normalization ofbranched circle patterns, it is proved that these approximating solutions convergeto F.