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 $n \geq 2$ and $c \in \mathbb{C}^*$. 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 $n \geq 2$, 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 $f_{n,-1}$.