Dimension of the Mandelbrot Set Boundary

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 → ∞.

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.

Return to Some features of the Mandelbrot set boundary.