Illinois Journal of Mathematics

Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension

David M. Freeman

Full-text: Open access

Abstract

We characterize fractal chordarc curves in Euclidean space by the fact that they remain bilipschitz homogeneous under inversion. We illustrate this result by constructing two examples. The techniques used in these constructions provide a means of calculating various dimensions of bilipschitz homogeneous Jordan curves.

Article information

Source
Illinois J. Math., Volume 54, Number 2 (2010), 753-770.

Dates
First available in Project Euclid: 14 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1318598680

Digital Object Identifier
doi:10.1215/ijm/1318598680

Mathematical Reviews number (MathSciNet)
MR2846481

Zentralblatt MATH identifier
0922.30017

Subjects
Primary: 30C35: General theory of conformal mappings
Secondary: 51F99: None of the above, but in this section

Citation

Freeman, David M. Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension. Illinois J. Math. 54 (2010), no. 2, 753--770. doi:10.1215/ijm/1318598680. https://projecteuclid.org/euclid.ijm/1318598680


Export citation

References

  • A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995, Corrected reprint of the 1983 original.
  • C. J. Bishop, Bi-Lipschitz homogeneous curves in $\Bbb R\sp 2$ are quasicircles, Trans. Amer. Math. Soc. 353 (2001), 2655–2663 (electronic).
  • D. M. Freeman and D. A. Herron, Bilipschitz homogeneity and inner diameter distance, to appear in J. Anal. Math.
  • D. M. Freeman, Unbounded bilipschitz homogeneous Jordan curves, submitted.
  • D. M. Freeman, Ph.D. thesis, Dept. of Mathematical Sciences, University of Cincinnati, 2009; available at http://etd.ohiolink.edu/view.cgi?acc_num=ucin1251229498.
  • M. Ghamsari and D. A. Herron, Higher dimensional Ahlfors regular sets and chordarc curves in $\bold R\sp n$, Rocky Mountain J. Math. 28 (1998), 191–222.
  • M. Ghamsari and D. A. Herron, Bi-Lipschitz homogeneous Jordan curves, Trans. Amer. Math. Soc. 351 (1999), 3197–3216.
  • D. A. Herron and V. Mayer, Bi-Lipschitz group actions and homogeneous Jordan curves, Illinois J. Math. 43 (1999), 770–792.
  • V. Mayer, Trajectoires de groupes à $1$-paramètre de quasi-isométries, Rev. Mat. Iberoamericana 11 (1995), 143–164.
  • S. Rohde, Quasicircles modulo bilipschitz maps, Rev. Mat. Iberoamericana 17 (2001), 643–659.