Self-similarity for the tricorn

TL;DR

This paper explores the self-similar structure of the tricorn, demonstrating the existence of many Mandelbrot set copies and showing the discontinuity of the straightening map for certain regions.

Contribution

It establishes the presence of numerous homeomorphic Mandelbrot copies within the tricorn and proves the straightening map's discontinuity in specific cases.

Findings

Existence of many homeomorphic Mandelbrot copies in the tricorn

Discontinuity of the straightening map for a baby tricorn at the airplane

Use of rigorous numerical methods to support theoretical results

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.