The proof of discontinuity of straightening maps is carried out by showing that all non-real umbilical cords of the multicorns wiggle, which settles a conjecture made by various people including Hubbard, Milnor, and Schleicher. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and antiholomorphic polynomials from their parabolic germs..
Source: wiktionary