On quasisymmetric mappings in semimetric spaces

Abstract

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.