On quasisymmetric mappings between ultrametric spaces

Abstract

In 1980 P. Tukia and J. V\"{a}is\"{a}l\"{a} in seminal paper [P. Tukia and J. V\"ais\"al\"a, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980)] extended a concept of quasisymmetric mapping known from the theory of quasiconformal mappings to the case of general metric spaces. They also found an estimation for the ratio of diameters of two subsets which are images of two bounded subsets of a metric space under a quasisymmetric mapping. We improve this estimation for the case of ultrametric spaces. It was also shown that the image of an ultrametric space under an $\eta$-quasisymmetric mapping with $\eta(1)=1$ is again an ultrametric space. In the case of finite ultrametric spaces it is proved that such mappings are ball-preserving.