The Multi-Valued Quasimöbius Mappings on the Riemann Sphere

Suppose that a multi-valued mapping of a domain in the sphere with disjoint images of distinct points boundedly distorts the Ptolemaic characteristic of generalized tetrads (quadruples of disjoint compact sets). Suppose that the image of each has at most components, each of which is a continuum of bounded turning. Then , up to the values at some isolated branch points, is the inverse of a mapping with bounded distortion in the sense of Reshetnyak. In particular, if then the left inverse to is the composition of a quasiconformal automorphism of and a rational function.