Newton-Like Components in the Chebyshev–Halley Family of Degree n Polynomials

Abstract

We study the Chebyshev–Halley methods applied to the family of polynomials f_{n,c}(z)=z^n+c, for nge 2 and cin mathbb {C}^{*}. We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for n ge 2, the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton’s method to f_{n,-1}.