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

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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 Möbius 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 Möbius property only in the class of univalent analytic functions f(z).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. V. Aseev

  2. Gorno-Altaĭsk State University, Gorno-Altaĭsk, Russia

    T. A. Kergilova

Authors
  1. V. V. Aseev
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  2. T. A. Kergilova
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Correspondence to V. V. Aseev.

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Original Russian Text Copyright © 2011 Aseev V. V. and Kergilova T. A.

The authors were supported by the Russian Foundation for Basic Research (Grants 09-01-98001-r-sibir-a and 11-01-90704-mob-st).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 5, pp. 977–992, September–October, 2011.

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Aseev, V.V., Kergilova, T.A. A four-point criterion for the Möbius property of a homeomorphism of plane domains. Sib Math J 52, 776–787 (2011). https://doi.org/10.1134/S0037446611050028

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