On McMullen-like mappings

Abstract

We introduce a generalization of particular dynamical behavior for rational maps. In 1988, C. McMullen showed that the Julia set of $f_{\lambda }(z)=z^n+\lambda /z^d$ for $|\lambda |\ne 0$ small enough is a Cantor set of circles if and only if $1/n+1/d<1$ holds. Several other specific singular perturbations of polynomials have been studied in recent years, all have parameter values where a Cantor set of circles is present in the associated Julia set. We unify these examples by defining a McMullen-like mapping as a rational map $f$ associated to a hyperbolic post critically finite polynomial $P$ and a pole data $\mathcal{D}$ where we encode the location of every pole of $f$ and the local degree at each pole. As for the McMullen family $f_{\lambda }$, we characterize a McMullen-like mapping using an arithmetic condition depending only on $(P,\mathcal{D})$. We show how to check the definition in practice providing new explicit examples of McMullen-like mappings for which a complete topological description of their Julia sets is made.