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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     author = {N. V. Abrosimov and V. V. Aseev},
     title = {Multivalued quasim\"obius property and bounded turning},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1185--1199},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a65/}
}
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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/