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

TL;DR

This paper investigates a new class of Sierpiński carpets, demonstrating their unique quasisymmetric properties and establishing conditions for their equivalence, revealing their rigidity and symmetry characteristics.

Contribution

It introduces the class $F_{n,p}$ of Sierpiński carpets, proves their quasisymmetric rigidity, and characterizes when two such carpets are quasisymmetrically equivalent.

Findings

The quasisymmetric self-maps of $F_{n,p}$ are exactly the Euclidean isometries.

$F_{n,p}$ and $F_{n',p'}$ are quasisymmetrically equivalent iff $(n,p)=(n',p')$.

The class $F_{n,p}$ is not quasisymmetrically equivalent to the standard Sierpiński carpet.

Abstract

We study a new class of square Sierpi\'nski carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpi\'{n}ski 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 $F_{n',p'}$ are quasisymmetrically equivalent if and only if $(n,p)=(n',p')$.