Geodesic
Multivalued quasim\"obius property and bounded turning
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1185-1199.

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The class of multivalued mappings with bounded angular distortion (BAD) property in metric spaces can be considered as a multivalued analogгу for quasimöbius mappings. We study the connections between quasimeromorphic self-mappings of $X= \bar{R}^n$ and multivalued mappings $F: X\to 2^X$ with BAD property. The main result of the paper concerns the multivalued mappings $F: D\to 2^{\bar{\mathbf C}}$ with BAD property of a domain $D\subset \bar{\mathbf{C}}$. If the image $F(x)$ of each point $x\in D$ is either a point or a continuum with bounded turning then $F$ is proved to be a single-valued quasimöbius mapping. The crucial point in the proof of this result is the local connectedness of the set $F(X)$ for the multivalued continuous mapping $F: X\to 2^Y$ with BAD property. We obtain sufficient conditions providing $F(X)$ to have local connectedness or bounded turning property in the most general case.
Keywords: multivalued quasimöbius mapping, multivalued hyperinjective mapping, Ptolemaic characteristic of tetrad, generalized angle, bounded angular distortion, local connectedness. 
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N. V. Abrosimov; V. V. Aseev. Multivalued quasim\"obius property and bounded turning. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1185-1199. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a65/

[1] P. Tukia, J. Väisälä, “Quasisymmetric embeddings of metric spaces”, Ann. Acad. Sci. Fenn., Ser. A I Math., 5:1 (1980), 96–114 | MR | Zbl

[2] J.Väisälä, “Quasi-symmetric embeddings in Euclidean spaces”, Trans. Am. Math. Soc., 264:1 (1981), 191–204 | MR | Zbl

[3] J.Väisälä, “Quasimöbius maps”, J. Anal. Math., 44 (1985), 218–234 | DOI | Zbl

[4] V.V. Aseev, A.V. Sychev, A.V. Tetenov, “Möbius-invariant metrics and generalized angles in Ptolemaic spaces”, Sib. Math. J., 46:2 (2005), 189–204 | DOI | MR | Zbl

[5] V.V. Aseev, “Generalized angles in Ptolemaic Möbius structures. II”, Sib. Math. J., 59:5 (2018), 768–777 | DOI | MR | Zbl

[6] V.V. Aseev, “Generalized angles in Ptolemaic Möbius structures”, Sib. Math. J., 59:2 (2018), 189–201 | DOI | MR | Zbl

[7] V.V. Aseev, “Multivalued mappings with quasimöbius property”, Sib. Math. J., 60:5 (2019), 741–756 | DOI | MR | Zbl

[8] V.V. Aseev, “On coordinate vector-functions of quasiregular mappings”, Sib. Èlectron. Math. Izv., 15 (2018), 768–772 | DOI | MR | Zbl

[9] V.V. Aseev, “Multivalued quasimöbius mappings of circle to circle”, Sib. Math. J., 62:1 (2021), 14–22 | DOI | MR | Zbl

[10] P. Tukia, “Spaces and arcs of bounded turning”, Mich. Math. J., 43:3 (1996), 559–584 | DOI | MR | Zbl

[11] K. Kuratowski, Topology, Academic Press and Polish Scientific Publishers, New York–London–Warszawa, 1966 | Zbl

[12] L.M. Blumenthal, Theory and applications of distance geometry, Clarendon Press, Oxford, 1953 | MR | Zbl

[13] M.H.A. Newman, Elements of the topology of plane sets of points, Cambridge Univ. Press, Cambridge, 1939 | MR | Zbl