Asymptotically symmetric embeddings and symmetric quasicircles

Brania, Abdelkrim; Yang, Shanshuang

doi:10.1090/S0002-9939-04-07375-7

Asymptotically symmetric embeddings and symmetric quasicircles
HTML articles powered by AMS MathViewer

by Abdelkrim Brania and Shanshuang Yang PDF

Proc. Amer. Math. Soc. 132 (2004), 2671-2678 Request permission

Abstract:

A well-known characterization of quasicircles is the following: A Jordan curve $J$ in the complex plane is a quasicircle if and only if it is the image of the unit circle under a quasisymmetric embedding. In this paper we try to characterize a subclass of quasicircles, namely, symmetric quasicircles, by introducing the concept of asymptotically symmetric embeddings. We show that a Jordan curve $J$ in the complex plane is a symmetric quasicircle if and only if it is the image of the unit circle under an asymptotically symmetric embedding.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
Jochen Becker and Christian Pommerenke, Über die quasikonforme Fortsetzung schlichter Funktionen, Math. Z. 161 (1978), no. 1, 69–80 (German). MR 507822, DOI 10.1007/BF01175613
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683–736. MR 1175689, DOI 10.2307/2374795
Frederick W. Gehring, Characterizations of quasidisks, Quasiconformal geometry and dynamics (Lublin, 1996) Banach Center Publ., vol. 48, Polish Acad. Sci. Inst. Math., Warsaw, 1999, pp. 11–41. MR 1709972
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
M. Kanarsky, On quasicircles and dilatations of quasisymmetric homeomorphisms, Ph.D. Thesis, Emory University, 2000.
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
Shengjian Wu and Shanshuang Yang, On symmetric quasicircles, J. Austral. Math. Soc. Ser. A 68 (2000), no. 1, 131–144. MR 1727222, DOI 10.1017/S1446788700001622