Square Sierpiski carpets and Lattes maps

摘要

We prove that every quasisymmetric homeomorphism of a standard square Sierpiski carpet Sp, p >= 3 odd, is an isometry. This strengthens and completes earlier work by the authors (Bonk and Merenkov in Ann Math (2) 177:591-643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of Sp across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet Sp is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.