  A small perturbation of a quadratic polynomial with a non-repelling fixed
point gives a polynomial with an attracting fixed point and a Jordan curve
Julia set, on which the perturbed polynomial acts like angle doubling. However,
there are cubic polynomials with a non-repelling fixed point, for which no
perturbation results into a polynomial with Jordan curve Julia set. Motivated
by the study of the closure of the Cubic Principal Hyperbolic Domain, we
describe such polynomials in terms of their quadratic-like restrictions.
