# The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

### Abstract

It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c\in \partial M$, the Julia set of $z\mapsto z^2 +c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

## Authors

Mitsuhiro Shishikura