# Hausdorff dimension of subsets of the Mandelbot set.

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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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. –  Alexis Monnerot-Dumaine Sep 8 '10 at 10:37