Geodesic
On quasisymmetric mappings in semimetric spaces
Evgeniy Petrov1 ; Ruslan Salimov2
1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
2 Institute of Mathematics of the NAS of Ukraine
Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 723-745.

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The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors in 1956. In 1980 Tukia and Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of a quasisymmetric mapping to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation "to lie between" are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we give a new estimate for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which form a special class of mappings between semimetric spaces.
DOI : 10.54330/afm.116845
Keywords: Quasisymmetric mapping, semimetric space, triangle function, weak similarity

Evgeniy Petrov 1 ; Ruslan Salimov 2

1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
2 Institute of Mathematics of the NAS of Ukraine 
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Evgeniy Petrov; Ruslan Salimov. On quasisymmetric mappings in semimetric spaces. Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 723-745. doi : 10.54330/afm.116845. http://geodesic.mathdoc.fr/articles/10.54330/afm.116845/

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