# 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).

revisions: 19930203 oldest on record; 20230616 definition in terms of neighborhood; reference McMullen

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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