JournalsjfgVol. 2, No. 3pp. 249–279

On McMullen-like mappings

  • Antonio Garijo

    Universitat Rovira i Virgili, Tarragona, Spain
  • Sébastien Godillon

    University de Barcelona, Spain
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Abstract

We introduce a generalization of particular dynamical behavior for rational maps. In 1988, C. McMullen showed that the Julia set of fλ(z)=zn+λ/zdf_{\lambda}(z)=z^n+\lambda/z^d for λ0|\lambda|\neq 0 small enough is a Cantor set of circles if and only if 1/n+1/d<11/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 ff associated to a hyperbolic post critically finite polynomial PP and a pole data D\mathcal D where we encode the location of every pole of ff and the local degree at each pole. As for the McMullen family fλf_{\lambda}, we characterize a McMullen-like mapping using an arithmetic condition depending only on (P,D)(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.

Cite this article

Antonio Garijo, Sébastien Godillon, On McMullen-like mappings. J. Fractal Geom. 2 (2015), no. 3, pp. 249–279

DOI 10.4171/JFG/21