On the inhomogeneity of the Mandelbrot set

We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\mathrm{\partial }M$ and satisfying $f(U\cap \mathrm{\partial }M)\subset \mathrm{\partial }M$ 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.