Shishikura's proof established the the long-standing "hairiness" conjecture. The proof is very subtle, but here is a rough outline of the steps.
In the cusp of the big cardioid, Shishikura found a sequence of Misiurewicz points cn with dim(Jcn) -> 2 as n -> infinity.
By Tan-Lei's theorem, regions in M around these cn have dim -> 2.
Every circle around every boundary point of the Mandelbrot set encloses copies of the Mandelbrot set, and the cusps of these copies have regions with dim -> 2.
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