Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Bounded turning circles are weak-quasicircles

Author: Daniel Meyer
Journal: Proc. Amer. Math. Soc. 139 (2011), 1751-1761
MSC (2010): Primary 30C65; Secondary 51F99
Published electronically: October 20, 2010
MathSciNet review: 2763763
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a metric Jordan curve $ \Gamma$ is bounded turning if and only if there exists a weak-quasisymmetric homeomorphism $ \varphi\colon \mathsf{S}^1 \to \Gamma$.

References [Enhancements On Off] (What's this?)

  • [AB35] F. Alt and G. Beer, Der $ n$-Gittersatz in Bogen, Ergebisse math. Koll. 6 (1935), 7.
  • [AB56] L. V. Ahlfors and A. Beurling, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142. MR 0086869 (19:258c)
  • [Ahl63] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. MR 0154978 (27:4921)
  • [Hei01] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • [HM] D. A. Herron and D. Meyer, Quasicircles and bounded turning circles modulo bi-Lipschitz maps, to appear in Rev. Mat. Iberoamericana.
  • [Men30] K. Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 103 (1930), no. 1, 466-501. MR 1512632
  • [Sch40] I. J. Schoenberg, On metric arcs of vanishing Menger curvature, Ann. of Math. (2) 41 (1940), 715-726. MR 0002903 (2:130b)
  • [TV80] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 5, 97-114. MR 595180 (82g:30038)
  • [Väi82] J. Väisälä, Dividing an arc to subarcs with equal chords, Colloq. Math. 46 (1982), no. 2, 203-204. MR 0678135 (84d:54056)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C65, 51F99

Retrieve articles in all journals with MSC (2010): 30C65, 51F99

Additional Information

Daniel Meyer
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 University of Helsinki, Finland

Keywords: Quasisymmetry, weak-quasisymmetry, bounded turning, weak-quasicircle.
Received by editor(s): March 30, 2010
Received by editor(s) in revised form: May 22, 2010
Published electronically: October 20, 2010
Additional Notes: The author’s research was supported by the Academy of Finland, projects SA-134757 and SA-118634
Communicated by: Mario Bonk
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society