On the Inhomogeneity of the Mandelbrot Set

Abstract

Abstract We will show that the Mandelbrot set $M$ is locally conformally inhomogeneous; the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap \partial M)=f(U)\cap \partial M$ 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 that the dynamics can be recovered from the local conformal structure of the Julia sets.