Square Sierpi\'nski carpets and Latt\`es maps

TL;DR

This paper proves that all quasisymmetric homeomorphisms of certain square Sierpiński carpets are isometries, and shows these carpets are not quasisymmetrically equivalent to Julia sets of postcritically-finite rational maps.

Contribution

The authors establish that quasisymmetric homeomorphisms of standard square Sierpiński carpets with odd p are isometries, extending previous results and applying techniques to related structures.

Findings

All quasisymmetric homeomorphisms of $S_p$ are isometries.

No standard square carpet $S_p$ is quasisymmetrically equivalent to a Julia set of a postcritically-finite rational map.

The methods apply to the double of $S_p$ across the outer circle.

Abstract

We prove that every quasisymmetric homeomorphism of a standard square Sierpi\'nski carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and completes earlier work by the authors. We also show that a similar conclusion holds for quasisymmetries of the double of $S_p$ across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet $S_p$ is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.