On locally distinguishing Sierpiński dynamical systems

Abstract

We prove that there is no local quasiconformal map between the limit set of a convex-cocompact Kleinian group and the Julia set of a postcritically finite rational map, provided that both are Sierpiński carpets. This contrasts with the recent results by Y. Luo, D. Ntalampekos and Y. Luo, M. Mj, S. Mukherjee for gasket and tree-like spaces, respectively.