Keywords: cross-ratio, linear-fractional map, extended complex plane, Borel measurability.
@article{VNGU_2010_10_4_a5,
author = {T. A. Kergilova},
title = {An injective {Borel} measurable map preserving fixed cross-ratio up to complex conjugation is a {M\"obius} map},
journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
pages = {68--81},
year = {2010},
volume = {10},
number = {4},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VNGU_2010_10_4_a5/}
}
TY - JOUR AU - T. A. Kergilova TI - An injective Borel measurable map preserving fixed cross-ratio up to complex conjugation is a Möbius map JO - Sibirskij žurnal čistoj i prikladnoj matematiki PY - 2010 SP - 68 EP - 81 VL - 10 IS - 4 UR - http://geodesic.mathdoc.fr/item/VNGU_2010_10_4_a5/ LA - ru ID - VNGU_2010_10_4_a5 ER -
%0 Journal Article %A T. A. Kergilova %T An injective Borel measurable map preserving fixed cross-ratio up to complex conjugation is a Möbius map %J Sibirskij žurnal čistoj i prikladnoj matematiki %D 2010 %P 68-81 %V 10 %N 4 %U http://geodesic.mathdoc.fr/item/VNGU_2010_10_4_a5/ %G ru %F VNGU_2010_10_4_a5
T. A. Kergilova. An injective Borel measurable map preserving fixed cross-ratio up to complex conjugation is a Möbius map. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 10 (2010) no. 4, pp. 68-81. http://geodesic.mathdoc.fr/item/VNGU_2010_10_4_a5/
[1] Alfors L., Preobrazovaniya Mebiusa v mnogomernom prostranstve, Mir, M., 1986 | MR
[2] Caratheodory C., “The Most General Transformations of Plane Regions which Transform Circles into Circles”, Bull. Amer. Math. Soc., 43 (1937), 573–579 | DOI | MR
[3] Benz W., “Characterization of Geometrical Mappings under Mild Hypothesis: Über ein Modernes Forschungsgebiet der Geometrie”, Hamb. Beitr. Wiss. gesch., 15 (1994), 393–409 | Zbl
[4] Kuzminykh A. V., “O edinichnykh bazakh evklidovoi metriki”, Sib. mat. zhurn., 38:4 (1997), 843–846 | MR
[5] Lester J. A., “Euclidean Plane Point-Transformations Preserving Unit Perimeter”, Arch. Math., 45 (1985), 561–564 | DOI | MR | Zbl
[6] Rassias Th. M., “Some Remarks on Isometric Mappings”, Facta Univ. Ser. Math. Inform., 2 (1987), 49–52 | MR | Zbl
[7] Khamsemanan N., Connelly R., “Two-Distance Preserving Functions”, Beitr. Algebra und Geom., 43:2 (2002), 557–564 | MR | Zbl
[8] Bogatyi S. I., Bogataya S. A., Frolkina O. D., “Affinnost otobrazhenii, sokhranyayuschikh ob'em”, Vestn. Mosk. gos. un-ta. Seriya 1. Mat., mekh., 2001, no. 6, 10–15 | MR
[9] Frolkina O. D., “Affinnost otobrazhenii, sokhranyayuschikh ugol”, Vestn. Mosk. gos. un-ta. Ceriya 1. Mat., mekh., 2002, no. 2, 60–63 | MR | Zbl
[10] Chubarev A., Pinelis I., “Fundamental Theorems of Geometry without the 1-to-1 Assumption”, Proc. Amer. Math. Soc., 127 (1999), 2735–2744 | DOI | MR | Zbl
[11] Yang S., Fang A., “A New Characteristic of Möbius Transformations in Hyperbolic Geometry”, J. Math. Anal. Appl., 319 (2006), 660–664 | DOI | MR | Zbl
[12] Li B., Wang Y., “A New Characterization for Isometries by Triangles”, New York J. Math., 15 (2009), 423–429 | MR | Zbl
[13] Haruki H., Rassias T. M., “A New Invariant Characteristic Property of Möbius Transformations from the Standpoint of Conformal Mapping”, J. Math. Anal. Appl., 181 (1994), 320–327 | DOI | MR | Zbl
[14] Haruki H., Rassias T. M., “A New Characteristic of Möbius Transformations by Use of Apollonius Points of Triangles”, J. Math. Anal. Appl., 197 (1996), 14–22 | DOI | MR | Zbl
[15] Haruki H., Rassias T. M., “A New Characteristic of Möbius Transformations by use of Apollonius Quadrilaterals”, Proc. Amer. Math. Soc., 126:10 (1998), 2857–2861 | DOI | MR | Zbl
[16] Haruki H., Rassias T. M., “A New Characterization of Möbius Transformations by Use of Apollonius Hexagons”, Proc. Amer. Math. Soc., 128 (2000), 2105–2109 | DOI | MR | Zbl
[17] Bulut S., Yilmaz Ozgur N., “A New Characteristic of Möbius Transformations by use of Apollonius Points of Pentagons”, Turkish J. Math., 28:3 (2004), 299–305 | MR | Zbl
[18] Bulut S., Yilmaz Ozgur N., “A New Characterization of Möbius Transformations by the Use of Apollonius Points of $(2n\,{-}\,1)$-gons”, Acta Mathematica Sinica. English Series, 21:3, June (2005), 667–672 | DOI | MR | Zbl
[19] Beardon A. F., Minda D., “Sphere-Preserving Maps in Inversive Geometry”, Proc. Amer. Math. Soc., 130 (2001), 987–998 | DOI | MR
[20] Kobayashi O., “Apollonius Points and Anharmonic Ratios”, Tokyo Math. J., 30:1 (2007), 117–119 | DOI | MR | Zbl
[21] Aseev V., Kergylova T., Moebius Transformations Preserving Fixed Anharmonic Ratio, 24 Oct. 2008, arXiv: 0810.4433v1[math.GT]
[22] Aczel J., McKiernan M. A., “On the Characterization of Plane Projective and Complex Möbius Transformation”, Math. Nachr., 33 (1967), 315–337 | DOI | MR | Zbl
[23] Hofer R., “A Characterization of Möbius Transformations”, Proc. Amer. Math. Soc., 128 (1999), 1197–1201 | DOI | MR
[24] Jing L., “A New Characterization of Möbius Transformations by use of Polygons Having Type $A$”, J. Math. Anal. Appl., 324 (2006), 281–284 | DOI | MR | Zbl
[25] Niamsup P., “A Characterization of Möbius Transformations”, Internat. J. Math. Math. Sci., 24:10 (2000), 663–666 | DOI | MR | Zbl
[26] Niamsup P., “A Note of the Characteristic of Möbius Transformations”, J. Math. Anal. Appl., 248 (2000), 203–215 | DOI | MR | Zbl
[27] Niamsup P., “A Note of the Characteristic of Möbius Transformations, II”, J. Math. Anal. Appl., 261 (2001), 151–158 | DOI | MR | Zbl
[28] Samaris Nikolas, “A New Characterization of Möbius Transformations by Use of $2n$ Points”, J. Nat. Geom., 22 (2002), 35–38 | MR | Zbl
[29] Ungar A., “The Hyperbolic Square and Möbius Transformations”, Banach J. Math. Anal., 1 (2007), 101–116 | MR | Zbl
[30] Yang S., “A Characterization of Möbius Transformations”, Proc. Japan Acad Ser. A Math. Sci., 84 (2008), 35–38 | DOI | MR | Zbl
[31] Federer G., Geometricheskaya teoriya mery, Nauka, M., 1987 | MR | Zbl
[32] Matematicheskaya entsiklopediya, v. 1, Izd-vo «Sovetskaya entsiklopediya», M., 1977
[33] Atsel Ya., Dombr Zh., Funktsionalnye uravneniya s neskolkimi peremennymi, Per. s angl., Fizmatlit, M., 2003