Hausdorff Dimensions Of Quasicircles

A k-quasicircle is the image of the unit circle under a k-quasiconformal mapping ofthe plane and a k-quasiline is the image of the real axis R under a k-quasiconformalmapping of the plane.In this thesis, we mainly study the relations between the Hausdorff dimensions ofk-quasicircles or k-quasilines and the theory of the universal Teichmuüller space. Thisthesis mainly contains four chapters.In chapter1, we introduce the definitions, notations as well as the basic theoryof quasiconformal mappings、Teichmuüller spaces and the Hausdorff dimensions ofquasicircles. Moreover, we give the outline of the problems discussed and the mainresults we have got.In chapter2, we study the relations between the Hausdorff dimensions of k-quasilines and the theory of extremal quasiconformal mappings. We show that there isan open and dense subset (Strebel points) of the universal Teichmuüller space T (H) suchthat, for every [f] in the set, the Hausdorff dimension of the k quasiline determinedby [f] is strictly less than1+k~2. We also show that there are some points [f]=[id]outside the open and dense set in the universal Teichmuüller space such that the Haus-dorff dimension of the quasiline determined by [f] is1. Moreover, Some results on theHausdorff dimensions of the quasilines varying in the asymptotic Teichmuüller space arealso obtained.In chapter3, we show that the Hausdorff dimensions of quasicircles of polygonalmappings is1. Furthermore, we apply this result to the theory of extremal quasicon-formal mappings. Let [μ] be a point in the universal Teichmuüller space such that theHausdorff dimension of the quasicircle f_μ(a△) is bigger than1. We show that for ev-ery knand polygonal differentials φ_n, n=1,2,···, the sequence {[k_nφ_n/|φ_n|]} cannotconverge to [μ] under the Teichmuüller metric.In the fourth chapter, we prove that, for any Fuchsian group Γ of the second kind and for any [μ] in the Teichmuüller space T (Γ), the Hausdorff dimension of the quasi-circle f_μ(a△) is not real analytic in the Teichmuüller space T (Γ).