Square Sierpi\'nski carpets and Latt\`es maps
Mario Bonk, Sergei Merenkov

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 , 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 across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.
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