Constructing Entire Functions by Quasiconformal Folding

We give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of the level-set {|f | > R}. The method consists of first building a quasiregular map by “gluing together” copies of the right half-plane that have each been quasiconformally “folded” into themselves. The measurable Riemann mapping theorem is then invoked to produce an entire function with similar geometry. Among other applications, we construct a function f with bounded singular set, whose Fatou set contains a wandering domain. Date: July 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: