Quasisymmetric rigidity of Sierpiński carpets

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 $(n,p)=(n^{\prime },p^{\prime })$.