On McMullen-like mappings

Abstract

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d<1$ and the simple critical values of $f_{\lambda}$ belong to the trap door. We generalize this behavior defining a McMullen-like mapping as a rational map $f$ associated to a hyperbolic postcritically finite polynomial $P$ and a pole data $\mathcal{D}$ where we encode, basically, the location of every pole of $f$ and the local degree at each pole. In the McMullen family, the polynomial $P$ is $z\mapsto z^n$ and the pole data $\mathcal{D}$ is the pole located at the origin that maps to infinity with local degree $d$. As in the McMullen family $f_{\lambda}$, we can characterize a McMullen-like mapping using an arithmetic condition depending only on the polynomial $P$ and the pole data $\mathcal{D}$. We prove that the arithmetic condition is necessary using the theory of Thurston's obstructions, and sufficient by quasiconformal surgery.