# Fatou Dust

Robert P. Munafo, 2002 Apr 22.

A Julia set that is a Cantor set is called a Fatou dust. Such a Julia set has an infinite number of pieces, equal in number to a continuum, each of which is a single point, and no two "touch" each other.

All Fatou Dusts have a parameter that is not in the Mandelbrot set. See also Fundamental Dichotomy.

All embedded Julia sets are shaped like Fatou dusts. The Fatou dust in an embedded Julia set is a limit set of the nodes, which are connected to the Julia set's center by a binary tree of branching filaments. This structure is particularly evident in the Type CC embedded Julia sets, which are shaped like cauliflower Fatou dusts.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.

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