Quasisymmetric rigidity of Sierpinski carpets $F_{n,p}$

Abstract

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\le n,1\le p<\frac{n}{2}-1$) 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. We also establish that $F_{n,p}$ and $F_{n^{\prime },p^{\prime }}$ are quasisymmetrically equivalent if and only if $(n,p)=(n^{\prime },p^{\prime })$.

24 pages, 5 figures. arXiv admin note: text overlap with arXiv:1102.3224 by other authors

References in corpus (1)

• Uniformization of Sierpiński carpets in the plane