Quasisymmetric rigidity of self-similar square carpets

Abstract

Let $S$ be a self-similar square carpet obtained by subdividing the unit square into $n^2$ congruent cells, deleting a nonempty collection of strictly interior open cells with pairwise disjoint closures, and repeating the same pattern with the same orientation in every retained cell. We prove that every quasisymmetric self-homeomorphism of $S$, equipped with the Euclidean metric, is a Euclidean isometry. The result holds for every integer $n\ge 3$ and imposes no symmetry condition on the deletion pattern. Consequently, the quasisymmetric and isometric groups coincide with the stabilizer of the first-generation holes in the symmetry group of the square. The proof combines round uniformization and Schottky-map stabilization with a propagation argument across retained cells. This produces global homothety relations for conjugates of cell contractions and forces preservation of the outer peripheral square. Periodic points on its sides then yield preservation of the corners, reducing the conclusion to the four-corner rigidity theorem for square carpets.