Geodesic
An injective Borel measurable map preserving fixed cross-ratio up to complex conjugation is a M\"obius map
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 10 (2010) no. 4, pp. 68-81

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The paper presents criterion for Möbius mappings in the complex plane: if an injective Borel measurable map preserves fixed cross-ratio $\lambda\notin \{0, 1, \infty\}$ up to complex conjugation, then it is a Möbius map.
Mots-clés : Moebius transformation, complex conjugation
Keywords: cross-ratio, linear-fractional map, extended complex plane, Borel measurability. 
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     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},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2010_10_4_a5/}
}
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T. A. Kergilova. An injective Borel measurable map preserving fixed cross-ratio up to complex conjugation is a M\"obius 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/