Circle homeomorphisms and shears

摘要

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation. The space Homeo(S~(1)) of orientation preserving homeomorphisms of the unit circle S~(1) is a classical topological group which is of interest in various fields of mathematics; see Ghys [11]. Its subgroup QS(S~(1)) of quasisymmetric maps of S~(1) plays a fundamental role in the Teichmuller theory of Riemann surfaces; see Ahlfors [2], Bers [3] and Gardiner and Lakic [9]. In fact, the universal Teichmuller space consists of all qua-sisymmetric maps which fix three distinguished points on S~(1) namely it is isomorphic to Mob(S~(1))QS(S~(1)), where Mob(S~(1)) is the group of (orientation preserving) Mobius maps which preserve S~(1) [3]. The subgroup Sym(S~(1)) of symmetric maps plays a prominent role in studying Teichmuller spaces of real dynamical systems; see Gardiner and Sullivan [10], Earle, Gardiner and Lakic [6] and Gardiner and Jiang [8].