# Multibrot Set

Robert P. Munafo, 2002 May 6.

The family of parameter-space sets (analogous to the Mandelbrot set) for the class of parametrized rational mappings (iteration functions) described by the equation:

Z' = Z^{p} + C

for some positive integer p>1. If p is 2, we get the normal Mandelbrot set; if p is higher, we get Mandelbrot-like sets with mu-atoms that have cusps, and with seeds that have one more cusp than their descendants.

(Some pictures are on Wikipedia: Multibrot set)

Multibrot sets have all of the same features and properties of the Mandelbrot set, including Julia sets, period scaling, miniature island multibrot sets, a simply-connected topology, all of the same types of symmetry, plus perfect (p-1)-fold rotational symmetry around the origin, embedded Julia sets, and the rest. Unlike most other mappings, the Multibrots do not have embedded Mandelbrot sets, because there is no place where their iteration dynamics has quadratic nature.

revisions: 20020506 oldest on record; 20230617 link to Wikipedia for pictures

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

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