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.
On measure and Hausdorff dimension of Julia sets for holomorphic Collet--Eckmann maps
Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set
Hausdorff dimension of divergent diagonal geodesics on product of finite volume hyperbolic spaces
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