On geometric definitions of quasisymmetric mappings

This paper focuses on the properties of quasisymmetric self–mappings of $$\mathbb {R}^n$$ with $$n\ge 2$$ . We obtain several geometric characterizations of quasisymmetric homeomorphisms in terms of ring domains and all pairs of disjoint continua by using conformal moduli and geometric moduli recently introduced by Tukia and Väisälä. As an application, we establish six necessary and sufficient conditions for a homeomorphism to be a quasimöbius transformation of the Riemann sphere $$\mathbb {\overline{R}}^n$$ onto itself.