Boundary of the Mandelbrot Set

Robert P. Munafo, 2023 Jun 16.

The boundary of the Mandelbrot set ∂M is the set of all points not in the interior, or equivalently, all points c for which any neighborhood no matter how small contains both interior and exterior points.

The boundary contains all of the chaotic behavior in the iteration algorithm: all points that iterate indefinitely without a period are in the boundary. All "interesting" views of the Mandelbrot Set contain points in the boundary.

The boundary can be mapped one-to-one onto a circle (see external angle), but at the same time it is infinitely convoluted, having a Hausdorff dimension of 2.0.

The boundary of the Mandelbrot Set is a fractal by Mandelbrot's definition, but not by the simple "dimension" definition since its dimension is 2.0. The Mandelbrot Set itself (boundary plus interior) is not a fractal by any definition.

Density of Islands

In the paper The Mandelbrot set is Universal Curtis T. McMullen states that the boundary of any "generalized Mandelbrot set" (which includes the normal Mandelbrot set) is a "bifurcation locus". When the generalization is for the family of polynomial iterations f={z^d+c} for all complex parameters c and some integer power d>1, called M_d (the degree d Multibrot set), small Mandelbrot sets (islands) are dense in the associated boundary ∂M_d. This seems to be equivalent to an open conjecture regarding islands in the boundary that would be equivalent to the unproven "MLC" local connectivity conjecture (see the open conjectures article); however it is subtly different because not all points in the boundary are of the well-behaved classes (such as Misiurewicz points).