A four-point criterion for the Möbius property of a homeomorphism of plane domains

Abstract

An ordered quadruple of pairwise distinct points T = {z 1, z 2, z 3, z 4} ⊂ C is called regular whenever z 2 and z 4 lie at the opposite sides of the line through z 1 and z 3. Consider Φ(T) = ∠z 1 z 2 z 3 + ∠z 1 z 4 z 3 (the angles are undirected) as some geometric characteristic of a regular tetrad. We prove the following theorem: For every fixed α ∈ (0, 2π) the Mobius property of a homeomorphism f: D → D* of domains in C is equivalent to the requirement that each regular tetrad T ⊂ D with Φ(T) = α whose image fT is also a regular tetrad satisfies Φ(fT) = α. In 1994 Haruki and Rassias established this criterion for the Mobius property only in the class of univalent analytic functions f(z).