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My second question about Shishikura's result :

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it has an area ? If yes, has anybody measured or calculated its "size" (Hausdorff measure) ? Thanks.

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(This used to be my research area, but I am not longer active in this topic, so I don't know all the latest references). Nevertheless, last year X. Buff and A. Chéritat (see the X. Buff's preprint page) proved that there exist Julia sets with positive Lebesgue measure (a result which was presented at this years' ICM), which would lend credence to the conjecture that so does the Mandelbrot set. But that, AFAIK, is still open. Xavier and Arnaud would be the best people from whom to ask this question.

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    $\begingroup$ The question is still open. I think most people still expect that the area is zero. Xavier's and Arnaud's work is an extremely important and beautiful breakthrough, but it is not quite clear whether it is relevant to the issue in question. The parameters considered belong to the boundaries of hyperbolic components, a set of Hausdorff dimension 1. (Their methods also yield "infinitely renormalizable polynomials" with positive measure Julia sets, which do not belong to the category I mention above, but again one would expect that the set of parameters produced is rather small.) $\endgroup$
    – Lasse Rempe
    Commented Nov 25, 2010 at 16:37
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This paper references an earlier paper which suggests (based on very indirect evidence from numerical estimates for the area of the interior) that the boundary may have positive Lebesgue measure. That was the most recent paper I could find on the subject so I tentatively infer that the answer to your question is that to date no-one has managed to calculate the area of the boundary.

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(Posting my old comment as an answer, as I probably should have done originally.)

Whether the boundary of the Mandelbrot set has positive area is a famous open problem, which has not been solved today.

(See e.g. the introduction of the paper "Collet, Eckmann and the bifurcation measure" by Astorg et al, where the question is referred to as "one of the important questions in complex dynamics which is still open".)

The general expectation is that the area is zero, but I would say there is not the same consensus on this as on the hyperbolicity conjecture or local connectivity of the Mandelbrot set.

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