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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Bi-Lipschitz homogeneous curves in $\mathbb{R} ^2$ are quasicircles


Author: Christopher J. Bishop
Journal: Trans. Amer. Math. Soc. 353 (2001), 2655-2663
MSC (2000): Primary 30C65
Published electronically: March 14, 2001
MathSciNet review: 1828465
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Abstract:

We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.


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Additional Information

Christopher J. Bishop
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: bishop@math.sunysb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02755-6
PII: S 0002-9947(01)02755-6
Keywords: Bi-Lipschitz mappings, homogeneous continua, quasicircles, bounded turning, quasihomogeneous embeddings, chord-arc, quasiconformal mappings, Hausdorff dimension
Received by editor(s): August 12, 1999
Published electronically: March 14, 2001
Additional Notes: The author is partially supported by NSF Grant DMS 98-00924
Article copyright: © Copyright 2001 American Mathematical Society