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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 [email protected], [email protected]
WEIXU SU
Affiliation:
School of Mathematics, Fudan University, 200433, Shanghai, PR China email [email protected], [email protected]
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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 

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