On the inhomogeneity of the Mandelbrot set

TL;DR

This paper proves that the Mandelbrot set exhibits local conformal inhomogeneity, meaning the only conformal self-maps fixing boundary points are the identity, highlighting its complex geometric structure.

Contribution

It establishes the inhomogeneity of the Mandelbrot set's boundary through analysis of local conformal symmetries and Julia set dynamics.

Findings

Mandelbrot set is locally conformally inhomogeneous.

Local conformal symmetries of Julia sets are often trivial.

Julia set dynamics can be recovered from local conformal structure.

Abstract

We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap\partial M)\subset \partial M$ is the identity map. The proof uses the study of local conformal symmetries of the Julia sets of polynomials: we will show in many cases, the dynamics can be recovered from the local conformal structure of the Julia sets.