Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T23:08:16.099Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasisymmetric rigidity of Sierpiński carpets $\boldsymbol{F}_{\boldsymbol{n},\boldsymbol{p}}$

Published online by Cambridge University Press:  04 June 2014

JINSONG ZENG  and
WEIXU SU
JINSONG ZENG
Affiliation:
School of Mathematics, Fudan University, 200433, Shanghai, PR China email 10110180006@fudan.edu.cn, suweixu@gmail.com
WEIXU SU
Affiliation:
School of Mathematics, Fudan University, 200433, Shanghai, PR China email 10110180006@fudan.edu.cn, suweixu@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n,1\leq p<(n/2)-1$) on $\mathbb{S}^{2}$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group of $F_{n,p}$. We also establish that $F_{n,p}$ and $F_{n^{\prime },p^{\prime }}$ are quasisymmetrically equivalent if and only if $(n,p)=(n^{\prime },p^{\prime })$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 5 , August 2015 , pp. 1658 - 1680
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlfors, L.. Lectures on Quasiconformal Mappings (Van Nostrand Mathematical Studies, 10). Van Nostrand, New York, 1966.Google Scholar
Ahlfors, L.. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973.Google Scholar
Bonk, M.. Quasiconformal geometry of fractals. Proc. Int. Congress in Mathematics (Madrid, 2006). European Mathematical Society, Zurich, 2006, pp. 1349–1373.Google Scholar
Bonk, M.. Uniformization of Sierpiński carpets in the plane. Invent. Math. 186 (2011), 559–665.CrossRefGoogle Scholar
Bonk, M., Kleiner, B. and Merenkov, S.. Rigidity of Schottky sets. Amer. J. Math. 131 (2009), 409–443.CrossRefGoogle Scholar
Bonk, M. and Merenkov, S.. Quasisymmetric rigidity of square Sierpiński carpets. Ann. of Math. (2) 177 (2013), 591–643.CrossRefGoogle Scholar
Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, 2001.CrossRefGoogle Scholar
Heinonen, J. and Koskela, P.. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181 (1998), 1–61.CrossRefGoogle Scholar
Kapovich, M. and Kleiner, B.. Hyperbolic groups with low-dimensional boundary. Ann. Sci. Éc. Norm. Supér (4) 33(5) (2000), 647–669.CrossRefGoogle Scholar
Lehto, O. and Virtanen, K. I.. Quasiconformal Mappings in the Plane, 2nd edn. Springer, New York–Heidelberg, 1973.CrossRefGoogle Scholar
Tukia, P. and Väisälä, J.. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Math. 5 (1980), 97–114.CrossRefGoogle Scholar
Whyburn, G. T.. Topological characterization of the Sierpiński curve. Fund. Math. 45 (1958), 320–324.CrossRefGoogle Scholar