Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


This paper is a survey of state of mathematical sciences related to the question on investigation of characterization of Möbius transformations under minimal assumptions.

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1. Beardon A. The geometry of discrete groups. Springer-Verlag. New York-Heidelberg-Berlin. (1983).

2. Ahlfors L.V. Möbius Transformations in Several Dimensions [Russian translation], Mir, Moscow (1986).

3. Carathéodory C. The most general transformations of plane regions which transform circles into circles. Bull. Amer. Math. Soc. Vol. 43. P. 537-579(1937).

4. Beardon A.F., Minda D. Sphere-preserving maps in inversive geometry. Proc. Amer. Math. Soc. Vol. 130, No 4. P. 987-998 (2001).

5. Cieśliński J. The cross ratio and Clifford algebras. Adv. in Clifford algebras. Vol. 7. No 2. P. 133-139 (1997).

6. Haruki H., Rassias Th. A new characterictic of Móbius transformations by use of Apollonius points of triangles. J. Math. Anal. Appl. Vol. 197. P. 14-22 (1996).

7. Kobayashi O. Apollonius points and anharmonic ratios. Tokyo Math. J. Vol. 30. No 1. P. 117-119 (2007).

8. Aseev V., Kergilova T. On transformations that preserve fixed anharmonic ratio Tokyo Math. J. Vol. 33. No 2. P. 365-371 (2010).

9. Aseev V.V., Kergilova T.A. Cross Ratio and Minimal Criteria for a Mapping to be Moebius. J. Math. Sci. 198 pp. 485–497 (2014).

10. Zelinskii Yu.B. On mappings invariant on subsets. Approximation Theory and Related Problems of Analysis and Topology [in Russian], Inst. Mat. AN UkrainSSR, Kiev, pp. 25–35 (1987).

11. Haruki H., Rassias Th. A new invariant characteristic property of Móbius transformations from standpoint of conformal mapping. J. Math. Anal. Appl. Vol. 181. P. 320-327 (1994).

12. 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 (2011).

13. Aseev V.V. Mappings slightly changing a fixed cross ratio. Sib. Math. J., 54, (2013).

14. Aseev V.V. A quasiconformal analogue of Caratheodory criterion for the Mobius property of mappings. Sib. Math. J. 55 (2014).

15. Aseev V.V. Quasiconformality of the injective mappings transforming spheres to quasispheres. Sib. Math. J. 57 (2016).

16. Höfer R. A characterization of Möbius transformations. Proc. Amer. Math. Soc. 128. No. 4 (1999).



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