Newton-like components in the Chebyshev-Halley family of degree $n$ polynomials

TL;DR

This paper investigates the dynamics of Chebyshev-Halley iterative methods applied to specific polynomial families, revealing complex basin structures and connections to Newton's method through quasiconformal deformations.

Contribution

It demonstrates the existence of parameters with infinitely connected basins and links the Julia sets of these methods to those of Newton's method via quasiconformal deformations.

Findings

Existence of parameters with infinitely connected basins of attraction.

Presence of a Julia set component as a quasiconformal deformation.

Connections between Chebyshev-Halley methods and Newton's method dynamics.

Abstract

We study the Cebyshev-Halley methods applied to the family of polynomials $f_{n,c}(z)=z^n+c$, for $n\ge 2$ 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 $n \ge 2$, 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 $f_{n,-1}$.