Square Sierpiński carpets and Lattès maps (Q2197669)

The authors are interested in the question of when two metric spaces are quasisymmetrically equivalent. In the paper under review, they study quasisymmetric homeomorphism of a standard square (planar) Sierpiński carpet \(S_p\). They prove that every quasisymmetric homeomorphism of \(S_p\), for \(p\geq 3\), is an isometry. This strengthens and completes earlier work by the authors [Ann. Math. (2) 177, No. 2, 591--643 (2013; Zbl 1269.30027)], where they proved that every quasisymmetry of \(S_3\) is an isometry (Theorem 1.1) and that the group of all of \(S_p\), \(p \geq 5\) and odd, is a finite dihedral group (Theorem 1.2). In the present paper they deal with this problem using new methods. Using the same methods, they show that a similar conclusion holds for quasisymmetries of the double of \(S_p\) across the outer peripheral circle. They also obtain applications in complex dynamics, proving in particular that no standard square carpet \(S_p\) is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.