Quasisymmetries of Sierpiński carpet Julia sets

Abstract

We prove that if $\xi$ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then $\xi$ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.