  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.
