Weakly Quasisymmetric Maps and Uniform Spaces

Suppose that X and Y are quasiconvex and complete metric spaces, that $$G\subset X$$G⊂X and $$G'\subset Y$$G′⊂Y are domains, and that $$f: G\rightarrow G'$$f:G→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 $$G'$$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 $$G'$$G′ is a uniform domain, then the images of the short arcs in G under f are uniform arcs in the sense of diameter.