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Asymptotically symmetric embeddings and symmetric quasicircles
Author(s):
Abdelkrim
Brania;
Shanshuang
Yang
Journal:
Proc. Amer. Math. Soc.
132
(2004),
2671-2678.
MSC (2000):
Primary 30C62
Posted:
March 25, 2004
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Abstract:
A well-known characterization of quasicircles is the following: A Jordan curve  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  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:
-
- [Ah]
- L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton, NJ, 1966. MR 34:336
- [AVV]
- G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, John Wiley and Sons, New York, 1997. MR 98h:30033
- [BP]
- J. Becker and C. Pommerenke, Über die quasikonforme Fortsetzung schlichter Funktionen, Math. Z. 161 (1978), 69-80. MR 58:22541
- [BA]
- A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142. MR 19:258c
- [GS]
- F. P. Gardiner and D. P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683-736. MR 95h:30020
- [Ge]
- F. W. Gehring, Characterizations of quasidisks, Polish Academy of Sciences, Warsaw, 1999, Banach Center Publications 48, 11-41. MR 2000g:30014
- [He]
- J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001. MR 2002c:30028
- [HK]
- J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 99j:30025
- [Ka]
- M. Kanarsky, On quasicircles and dilatations of quasisymmetric homeomorphisms, Ph.D. Thesis, Emory University, 2000.
- [Le]
- O. Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, New York, 1987. MR 88f:30073
- [Po]
- C. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992. MR 95b:30008
- [TV]
- P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. MR 82g:30038
- [WY]
- S. Wu and S. Yang, On symmetric quasicircles, J. Austral. Math. Soc. Ser. A 68 (2000), 131-144. MR 2000k:30027
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Additional Information:
Abdelkrim
Brania
Affiliation:
Department of Mathematics, Morehouse College, Atlanta, Georgia 30314
Email:
abrania@morehouse.edu
Shanshuang
Yang
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email:
syang@mathcs.emory.edu
DOI:
10.1090/S0002-9939-04-07375-7
PII:
S 0002-9939(04)07375-7
Keywords:
Quasisymmetric maps,
quasicircles,
asymptotically symmetric maps,
symmetric quasicircles
Received by editor(s):
September 18, 2002
Received by editor(s) in revised form:
June 12, 2003
Posted:
March 25, 2004
Communicated by:
Juha M. Heinonen
Copyright of article:
Copyright
2004,
American Mathematical Society
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