Quasisymmetric rigidity of square Sierpinski carpets

TL;DR

This paper proves that quasisymmetric self-maps of the standard 1/3-Sierpiński carpet are Euclidean isometries, and characterizes the quasisymmetric groups for a family of carpets, using a new discrete modulus invariant.

Contribution

It establishes the rigidity of quasisymmetric self-maps for the standard Sierpiński carpet and describes the symmetry groups for generalized carpets, introducing a novel invariant.

Findings

Quasisymmetric self-maps of the standard 1/3-Sierpiński carpet are Euclidean isometries.

The groups of quasisymmetric self-maps for generalized carpets are finite dihedral.

Different p-values yield non-quasisymmetrically equivalent carpets.

Abstract

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant---a certain discrete modulus of a path family---that is preserved under quasisymmetric maps of carpets.