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Newton-like components in the Chebyshev-Halley family of degree $n$ polynomials (2202.08787v1)

Published 17 Feb 2022 in math.DS and math.CV

Abstract: We study the Cebyshev-Halley methods applied to the family of polynomials $f_{n,c}(z)=zn+c$, 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 $f_{n,-1}$.

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