A THEOREM ON HOMEOMORPHISMS FOR ELLIPTIC SYSTEMS AND ITS APPLICATIONS

This paper is devoted to the substantiation of a criterion for the quasisymmetric conjugacy of an arbitrary group of homeomorphisms of the real line to a group of affine transformations (the Ahlfors problem). In a criterion suggested by Hinkkanen the constants in the definition of a quasisymmetric homeomorphism were assumed to be uniformly bounded for all elements of the group. Subsequently, for orientation-preserving groups this author put forward a more relaxed criterion, in which one assumes only the uniform boundedness of constants for each cyclic subgroup. In the present paper this relaxed criterion is proved for an arbitrary group of line homeomorphisms, which do not necessarily preserve the orientation. Bibliography: 4 titles. Introduction A homeomorphism g : R → R is said to be quasisymmetric [1] if it satisfies the condition M−1 g g(x+ t)− g(x) g(x) − g(x− t) Mg . (1) If g is a quasisymmetric homeomorphism, then the homeomorphism g−1 is also quasisymmetric. For arbitrary quasisymmetric homomorphisms g1 and g2 their composite is also a quasisymmetric homeomorphism and Mg1g2 Mg1Mg2 [1]. Since the constant Mg in condition (1) for a homeomorphism g is not unique, this means that there exists a constant Mg1g2 for the homeomorphism g1g2 such that the inequality holds. We say that a group G consisting of quasisymmetric homeomorphisms is quasisymmetric. The following basic result for a quasisymmetric group of line homeomorphisms was obtained in [2]. Theorem 1. Let G be a group of line homeomorphisms. Then a quasisymmetric homeomorphism η such that η ◦G ◦ η−1 is a group of affine transformations exists if and only if G is a quasisymmetric group such that Mg = M for all g ∈ G, where M is a fixed constant. This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 53-01-00174) and the Programme of Support of Leading Scientific Schools of RF (grant no. NSh-457.2003.1). AMS 2000 Mathematics Subject Classification. Primary 54H15; Secondary 20F38, 28D99.