摘要
我们称单位圆周(?)△在一个全平面的k拟共形映射下的像为一条k拟圆周；类似的,我们称实数轴R在一个全平面的k拟共形映射下的像为一条k拟线.在本学位论文中,我们主要考虑拟圆或者拟线的Hausdorff维数与万有Teichmuller空间之间的相关关系的一些问题.本学位论文共分四章.
第一章,我们主要介绍拟共形映射及Teichmuller理论和拟圆周的Hausdorff维数的一些基本定义和结果,以及我们得到的主要结果。
第二章,我们主要研究k拟线的Hausdorff维数与极值拟共形映射理论的关系.我们证明了在万有Teichmuller空间T(H)中存在一个开稠的子集(Strebel点集)E,使得E中的任何元素[f]所确定的k拟线的Hausdorff维数都不能达到1+k2.我们同时证明了开集E在T(H)的补集中也存在点[f]≠[id],使得其确定的拟线的Hausdorff维数甚至为1.在这章中,我们进一步的给出拟线的Hausdorff维数在渐进Teichmuller空间中变化规律的一些结果.
第三章,我们主要证明多边形映射所对应拟圆周的Hausdorff维数为1,并给出其在极值拟共形映射理论中的一些应用.设[μ]为万有Teichmuller空间T(△)中的一点,使得拟圆周fμ((?)△)的Hausdorff维数大于1.我们证明了对任意的kn∈(0,1)和多边形微分ψn,n=1,2,…,序列{[knψn/|ψn|]}在Teichmuller度量下不能收敛到[μ].
第四章,我们将证明,对于任意的第二类Fuchs群Γ,由[μ]∈T(Γ)所对应的拟圆周fμ((?)△)的Hausdorff维数确定的映射在Teichmuller空间T(Γ)中不实解析.
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).
引文
[1] W. Abikoff: The real analytic theory of Teichmu¨ller space. Lectures notes in Math-ematics, New York, Vol.820,(1986).
[2] L.V. Ahlfors: On Quasiconformal mappings, J. d’Analyse Math.,3,(1953),1-58.
[3] L.V. Ahlfors: The complex analytic structure of the space of closed Riemann sur-faces, R.Nevanlinna et al., Analytic Functions, Princeton Univ. Press, Princeton,N.J.1960,45-66.
[4] L.V. Ahlfors: Quasiconformal reflections, Acta Math.,109,(1963),291-301.
[5] L.V. Ahlfors: Lecture on quasiconformal mappings, Second Edition. Univ. Lec-ture.series. Vo.38. Am.Math.Soc.(2006).
[6] L.V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann.Math.72,(1960),385-404.
[7] S. Agard. A geometry proof of Mostow rigidity’s theorem for group of divergence,Acta. Math.,151,(1983),231-252.
[8] K. Astala: Area distortion of quasiconformal mappings, Acta Math., no.1,173,(1994),37-60.
[9] K. Astala, T. Iwaniec and G. Martin: Elliptic partial difential equations and qua-siconformal mappings in the plane, volume48of Princeton Mathematical series.Princeton University Press, Princeton,NJ,2009.
[10] K. Astala and M. Zinsmeister: Mostow rigidity and Fuchsian groups,C.R.Acad.sci.Paris.Ser., I311,(1990),301-306.
[11] K. Astala and M. Zinsmeister: Teichmu¨ller spaces and BMOA, Math. Ann.,289,(1991),613-625.
[12] K. Astala and M. Zinsmeister: Holomorphic families of quasi-Fuchsian groups,Ergod.Th and Dynam.sys.,14,(1994),207-212.
[13] C.J. Bishop. A criterion for failure of Rueller’s property, Ergod. Th. and Dy-nam.Syst.,26,(2006),1733-1748.
[14] C.J. Bishop: Divergence groups have the Bowen property, Annals of Math.,154,(2001),205-217
[15] L. Bers: A non-standard integral equation with applications to quasiconformalmappings, Acta. Math.,116,(1966),37-60.
[16] R. Bowen: Hausdorff dimension of quasicircles. Publ. Math. IHES,1250,(1979),11-25.
[17] A. Beurling and L.V.Ahlfors: The boundary correspondence under quasiconfor-mal mappings, Acta. Math.,96,(1956),125-142.
[18] J. Becker and C. Pommerenke: On the Hausdorff dimension of quasicircles, An-n.Acad.Sci.Fenn.Ser. A I Math.,12,(1987),329-333.
[19] G.Z. Cui and Y. Qi: Local boundary dilatation of quasiconformal maps in thedisk. Proc. Amer.Math.Soc.,Volume130, no5,(2001),1383-1389.
[20] G.Z. Cui and M. Zinsmeister: BMO-Teichmu¨ller spaces, Illinois Journal of Math.,48,(2004),1223-1233.
[21] A. Douady and C.J. Earle: Conformally natural extension of homeomorphisms ofthe circle, Acta Math.,157,(1986),23-48.
[22] G.A. Edgar: Classics on fractals, Addison-Wesley Publishing Company,1993.
[23] C.J. Earle, F.P. Gardiner and N. Lakic: Asymptotic Teichmu¨ller space I: The com-plex structure, In Contemp.Math.256,(2000),17-38.
[24] C.J. Earle, and Z. Li: Isometrically embedded polydisks in infinite dimensionalTeichmu¨ller spaces, The Journal of Geometric Analysis.9(No1),(1999),51-71.
[25] C.J. Earle, V. Markonic and Saric: Barycentric extension and the Bers embed-ding for asymptotic Teichmu¨ller space, Complex manifolds and hyperbolic ge-ometry (Guanajuato,2001),87-105, Contemp. Math.,311, Amer. Math. Soc. Prov-idence.RI,2002. MR1940165(3003i:30072).
[26] R. Fehlmann: Quasiconformal mappings with free boundary components, An-n.Acad.Sci.Fenn. Math.,72,(1982),337-347.
[27] F.P. Gardiner: Teichmu¨ller theory and quadratic differentials, John Wiley andSons, Inc., New York,1987. MR903027(88m:32044).
[28] F.P. Gardinerand N. Lakic: Quasiconformal Teichmu¨ller theory, Math SurveysMonogr.,76, Amer.Math. Soc.,Providence,RI2000. MR1730906(2001d:32016).
[29] J. Garnett. Bounded Analytic Functions, Academic:New York1981.
[30] J. Graczyk and P. Jones: Dimension of the boundary of quasiconformal Siegeldisk, Invent Math,148,(2002),465-493
[31] F.P. Gardiner and D.P. Sullivan: Symmetric structures on a closed curve, Amer. J.Math.,114,(1992),683-736.
[32] R. S. Hamilton: Extremal quasiconformal mappings with prescribed boundaryvalues, Trans. Amer. Soc.,138,(1969),399-406.
[33] Y. Imayoshi and M. Taniguchi: An introduction to Teichmu¨ller spaces, Springer-Verlag, Tokyo,1992.
[34] S. Krushkal: Teichmu¨ller theorem on extremal quasiconformal mappings, Siberi-an Math. J.2,(1967),231-244.
[35] S. Krushkal: Extremal quasiconformal mappings, Siberian Math. J.10,(1969),411-418.
[36] N. Lakic: The Strebel points, Contemp.Math.,211:417-431.
[37] O. Lehto: Univalent functions and Teichmu¨ller spaces,-Springer-Verlag, NewYork,1987.
[38] O. Lehto and K.I. Virtanen: Quasiconformal mappings in the plane, Springer-Verlag, New York,1973.
[39]李忠,拟共形映射及其在黎曼曲面中的应用,科学出版社,北京,(1988)
[40]李忠,复分析导引,北京大学出版社,(2004)
[41] Z. Li: Nonuniqueness of geodesics in infinite dimensional Teichmu¨ller spaces,Complex Variables Theory Appl.,16,(1991),261-272.
[42] C.T. Mcmullen: Hausdorff dimension and conformal dynamics I: Strong con-vergence of Kleinians groups,J. Differential Geom. Volume51, Number3(1999),471-515.
[43] C.T. Mcmullen: Hausdorff dimension and conformal dynamics II: Geometricallyfinite rational maps, Comm. Math. Helv.,75,(2000),535-593.
[44] C.T. Mcmullen: Hausdorff dimension and conformal dynamics III: Computationof Dimension, Am. J. Math.120, No4(1998),691-721.
[45] S. Nag: The Complex Analytic Theory of Teichmu¨ller space, Wiley-Interscience,New York (1988).
[46] S.J. Patterson: The limit of a Fuchsian group, Acta Math.,136,(1976),241-273.
[47] C. Pommerenke: Schlichte Funktionen und BMOA, Comment. Math. Helv.,52,(1977),591-602.
[48] C. Pommerenke: Polymorphic functions for group of divergence type, Math. Ann.,258,(1981),353-366.
[49] C. Pfluger: Quasikonforme Abbildungen und logarithmische Kapazitat, Ann. Inst.Feu. Grenoble.,2,(1951),69-80.
[50] C. Pfluger:U¨ber die konstruktion Riemannscher fla¨chen verheftung, Jour. Ind.Math. soc.,24,(1961),401-412.
[51] I. Prause: A remark on quasiconformal dimension distortion on the line, An-n.Acad.Sci.Fenn. Math.,32,(2007),341-352.
[52] D. Ruelle: Repellors for real analytic maps, Ergod.th. and Dynam.sys.,2,(1982),99-107.
[53] E. Reich and K. Strebel: On quasiconformal mappings which keep the boundarypoints fixed, Trans. Amer. Math. Soc.,138,(1969),211-222.
[54] E. Reich and K. Strebel: On the extremality of certain Teichmu¨ller mappings,Comment. Math. Helv.,45,(1970),353-362.
[55] E. Reich and K. Strebel: Extremal quasiconformal mappings with prescribedboundary values, In: Contributions to Analysis, A collection of papers dedicated toLipman Bers. New York: Academic Press,1974,375–391
[56] S. Smirnov: Dimension of quasicircles, Acta Math.,205,(2010),189-197.
[57] Y.L. Shen: A note on hamilton sequences for extremal Beltrami coefficients, ProcA M S.,129,(2000),105-109
[58] K. Strebel: Zur frage der Eindentigkeit extremaler quasikoformer Abbildungendes Einheitskreises. Comment. Math. Helv.,36,(1962),306-323.
[59] K. Strebel: On the existence of extremal Teichmu¨ller mappings. J. Anal. Math.,30,(1976),464-480.
[60] K. Strebel: Extremal Quasiconformal Mappings, Results in Mathematics.,10,(1986),168-210.
[61] D. Sullivan:The density at infinity of discrete of group of hyperbolic motions. Inst.Hautes. Etudes. Sci. Publ. Math.,50,(1979),419-450.
[62] D. Sullivan:Discrete of conformal groups and measurable dynamics. Bull. of AM S.,6,(1982),57-73.
[63] D. Sullivan: Hausdorff measures old and new, and the limit sets of geometricallyfinite Kleinian groups, Acta. Math.,153,(1984),259-277.
[64] D. Sullivan: Quasiconformal homoemorphisms and dynamics I: Solution of theFatou-Julia problem on wandering domain, Ann. Math.,122,(1985),401-418.
[65] D. Sullivan: Quasiconformal homoemorphisms and dynamics II: Structural sta-bility implies hyperbolicity for Kleinian groups, Acta. Math.,150,(1985),243-260.
[66] S.J. Wu: Moduli of quadrilaterals and extremal quasiconformal extensions of qua-sisymmetric functions, Comment Math. Helv.,72,(1997),593-604.
[67] S.J. Wu: On the Teichmu¨ller theorem and the heights theorem for quadratic dif-ferentials, Pro. Amer. Math. Soc.,129,(2001),765-770.
[68] M. Zinsmeister: Les Domaines de Carleson, Mich.Math. J.,36,(1989),213-220.