fractals - Hausdorff dimension of subsets of the Mandelbot set.

Yes - and it is indeed in Shishikura's paper, as in your comment.

The "open subsets" of the boundary with respect to which topology? That of the ambient space, or the relative topology of the set itself? Since the latter topology is extremely wild, it is much clearer to use the ambient topology, as Shishikura does.