Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations

The aim of this paper is to delve into the dynamic study of the well-known Chebyshev–Halley family of iterative methods for solving nonlinear equations. Our objectives are twofold: On the one hand, we are interested in characterizing the existence of extraneous attracting fixed points when the methods in the family are applied to polynomial equations. On the other hand, we are also interested in studying the free critical points of the methods in the family, as a previous step to determine the existence of attracting cycles. In both cases, we want to identify situations where the methods in the family have bad behavior from the root-finding point of view. Finally, and joining these two studies, we look for polynomials for which there are methods in the family where these two situations happen simultaneously. The rational map obtained by applying a method in the Chebyshev–Halley family to a polynomial has both super-attracting extraneous fixed points and super-attracting cycles different from the roots of the polynomial.