The Hausdorff dimension of the boundary of the Mandelbrot set and Julia   sets
  It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, 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. 
Mitsuhiro Shishikura