Dynamics of a family of Chebyshev-Halley-type methods

In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This cat set has interesting similarities with the Mandelbrot set. The parameters space has allowed us to find different elements of the family such that can not converge to any root of the polynomial, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.