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 [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.

KEYWORDS:

• anti-holomorphic dynamics
• complex dynamics
• computer-assisted proof

2010 AMS SUBJECT CLASSIFICATION:

• Primary 37F45
• Secondary 37F20
• 37F25

Acknowledgments

The author would like to thank Shigehiro Ushiki and Zin Arai for helpful discussions and comments. He would also like to thank Wolf Jung and Sabyasachi Mukherjee for helpful comments.

Additional information

Funding

Partially supported by JSPS KAKENHI Grant Numbers 22740105 and 26400115.