Hiroyuki Inou

In [M3], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci, and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.

It is well known that baby Mandelbrot sets are homeomorphic to the original one. We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically natural straightening map from a baby Tricorn to the original Tricorn is discontinuous at infinitely many explicit parameters. This is the first known example of discontinuity of straightening maps on a real two-dimensional slice of an analytic family of holomorphic polynomials. The proof of discontinuity is carried out by showing that all non-real umbilical cords of the Tricorn wiggle, which settles a conjecture made by various people including Hubbard, Milnor, and Schleicher.

We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.

We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author [IK 12], we show that there are many homeomorphic copies of the Mandelbrot sets. With the help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a “baby tricorn” centered at the airplane, hence not a homeomorphism.

We prove that if two globally-defined one-dimensional complex dynamics are locally analytically conjugate, then we extend the conjugacy to obtain global conjugacy by a correspondence. The most important case occurs when two rational maps have analytically conjugate polynomial-like restrictions. In this case, we prove that there exists another rational map which is semiconjugate to them both by some rational maps.