Self-similarity for the tricorn

Abstract

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, we show that there are many homeomorphic copies of the Mandelbrot sets. With 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.