On the inhomogeneity of the Mandelbrot set

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.

References in corpus (3)

• Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture
• On Ritt's polynomial decomposition theorems
• On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

Cited by in corpus (1)

• Geometry of PCF parameters in spaces of quadratic polynomials