Quasisymmetric rigidity of square Sierpiński carpets

Abstract

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpiński 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.