On quasisymmetric mappings between ultrametric spaces

TL;DR

This paper improves the understanding of quasisymmetric mappings in ultrametric spaces, showing they preserve ultrametric structure and are ball-preserving in finite cases, extending classical results from metric space theory.

Contribution

It refines diameter ratio estimates for quasisymmetric maps on ultrametric spaces and proves these mappings preserve ultrametricity and balls in finite cases.

Findings

Improved diameter ratio estimation for ultrametric spaces

Ultrametric spaces remain ultrametric under certain quasisymmetric maps

Finite ultrametric spaces have ball-preserving quasisymmetric mappings

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.