# Weakly quasisymmetric maps and uniform spaces

## TL;DR

This paper investigates the properties of weakly quasisymmetric maps between quasiconvex, complete metric spaces, showing that such maps preserve the uniformity of subdomains and the uniformity of short arcs under certain conditions.

## Contribution

It establishes that weakly quasisymmetric maps preserve the uniformity of subdomains and short arcs in quasiconvex, complete metric spaces, extending understanding of these mappings.

## Key findings

- Images of uniform subdomains are uniform under weakly quasisymmetric maps.
- Short arcs in the domain map to uniform arcs in the codomain.
- Weakly quasisymmetric maps preserve the uniformity of arcs in uniform domains.

## Abstract

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we show that: if $f$ is a weakly quasisymmetric mapping and $G'$ is a quasiconvex domain, then the image $f(D)$ of every uniform subdomain $D$ in $G$ is uniform. As an application, we get that if $f$ is a weakly quasisymmetric mapping and $G'$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter.