We introduce a generalization of particular dynamical behavior for rational maps. In 1988, C. McMullen showed that the Julia set of for small enough is a Cantor set of circles if and only if 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 associated to a hyperbolic post critically finite polynomial and a pole data where we encode the location of every pole of and the local degree at each pole. As for the McMullen family , we characterize a McMullen-like mapping using an arithmetic condition depending only on . 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.
Cite this article
Antonio Garijo, Sébastien Godillon, On McMullen-like mappings. J. Fractal Geom. 2 (2015), no. 3, pp. 249–279DOI 10.4171/JFG/21