Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials

Abstract

We study the Chebyshev-Halley family of root finding algorithms from the point of view of holomorphic dynamics. In this paper we provide a criterion which guarantees the simple connectivity of the basins of attraction of the roots. We use the criterion for the Chebyshev-Halley methods applied to the degree n polynomials zn+c, obtaining a characterization of the parameters for which all Fatou components are simply connected and, therefore, the Julia set is connected. We also study how increasing n affects the dynamics.