Rigidity of McMullen Julia sets

Abstract

We provide a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps $f_λ(z)=z^n+λ/z^n$ with $λ\in\mathbb{C}^*$ and $n\geq 2$, and prove that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results establish quasisymmetric rigidity for all topological classes in this family, including Sierpiński-like carpets, necklaces, and clusters, and provide the first known examples of rigid Julia sets in each of the three classes.