Abstract. In this paper, first, we define the weakly quasimöbius maps in quasi-metric spaces and obtain a series of elementary properties of these maps. Then we find conditions under which a weakly quasimöbius map is quasimöbius in quasi-metric spaces. With the aid of uniform perfectness, three related results are proved, and some applications are also given.

In this paper, we investigate the relationship between semisolidity and locally weak quasisymmetry of homeomorphisms in quasiconvex and complete metric spaces. Our main objectives are to (1) generalize the main result in [14] together with other related results, and (2) give a complete answer to the open problem given in [14]. As an application, we prove that the composition of two locally weakly quasisymmetric mappings is a locally weakly quasisymmetric mapping and that it is quasiconformal.2000 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.

In this paper, we establish several characterizations for quasihyperbolic mappings in Banach spaces. As an application, we provide a partial solution to a problem left open by Väisälä regarding the local properties of quasihyperbolic mappings.

The main purpose of this paper is to study the characterizations of John spaces. We obtain five equivalent characterizations for length John spaces. As an application, we establish a dimension-free quasisymmetric invariance of length John spaces. This result is new also in the case of the Euclidean space.2010 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.

Suppose that X and Y are quasiconvex and complete metric spaces, that G ⊂ X and G ′ ⊂ Y are domains, and that 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 ′ 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.