Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper, but I can't figure out one thing : can we say all open subsets of this boundary has dimension 2 ? Are there any references ? Thanks.
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asked Aug 25, 2010 at 14:12
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$\begingroup$ After all, the answer may be in the paper itself. Shishikura says : "Theorem A. H-dim(∂M) = 2. Moreover for any open set U which intersects ∂M, we have H-dim(∂M ∩ U) = 2." The way it is written seems strange to me. Why use this open set intersecting the boundary ? Does this answer the question ? Thanks. $\endgroup$– Alexis Monnerot-DumaineCommented Sep 8, 2010 at 10:37
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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.
answered Sep 9, 2010 at 23:44
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$\begingroup$ If S is an open subset of topological space T, then aren't open subsets of S in the subspace topology precisely open subsets of T intersected with S? Is there another topology being considered here besides the subspace topology? $\endgroup$ Commented Sep 10, 2010 at 2:16
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$\begingroup$ I think I had 'connected components' on the brain when I wrote that, rather than just considering the open subsets per se. $\endgroup$ Commented Sep 10, 2010 at 3:08