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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 theexistence of parameters such that the immediate basins of attractioncorresponding to the roots of unity are infinitely connected. We also provethat, for $n \ge 2$, the corresponding dynamical plane contains a connectedcomponent of the Julia set, which is a quasiconformal deformation of the Juliaset of the map obtained by applying Newton's method to $f_{n,-1}$.

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