On Uniformity Exponents of \(\varphi \)-Uniform Domains

Let \(G\subsetneq {\mathbb {R}}^n\) $G⊊{\mathbb{R}}^n$ be a domain, where \(n\ge 2\) $n\ge 2$ . Let \(k_G\) $k_G$ and \(j_G\) $j_G$ be the quasihyperbolic metric and the distance ratio metric on G, respectively. In the present paper, we prove that the identity map of \((G,k_G)\) $(G,k_G)$ onto \((G,j_G)\) $(G,j_G)$ is quasisymmetric if and only if it is bilipschitz. To classify domains of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ into various types according to the behaviors of their quasihyperbolic metrics, we define a uniformity exponent for every proper subdomain of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ and prove that this exponent may assume any value in \(\{0\}\cup [1,\infty ]\) $\{0\}\cup [1,\infty ]$ . Moreover, we study the properties of domains of uniformity exponent 1 and show by an example that such a domain may be neither quasiconvex nor accessible.