Connected McMullen-like Julia sets in a Chebyshev-Halley Family

Abstract

In this paper we study a one parameter family of rational maps obtained by applying the Chebyshev-Halley root finding algorithms. We show that the dynamics near parameters where the family presents some degeneracy might be understood from the point of view of singular perturbations. More precisely, we relate the dynamics of those maps with the one of the McMullen family $M_{\lambda}(z)=z^4 + \lambda /z^2$, using quasi-conformal surgery.