  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.
