Robert P. Munafo, 2002 Dec 5.
General definition: A Julia set is a maximal set of points (on the complex plane) with the property that any member of the set will be replaced with some other member of the set (or perhaps the same value) when mapped by a function Z' = f(Z). Generally the function f includes some constant parameter, and if so, the parameters with certain properties can be said to belong to the "Mandelbrot set" of f.
Specific to the "standard" Mandelbrot set: A Julia set is the maximal set of points that gets mapped onto itself under the function Z' = Z2 + C, for some constant C.
Some Julia Sets are connected, and others are disconnected; the disconnected Julia Sets are all Cantor sets (and are called Fatou dusts); see fundamental dichotomy. The topology of the Julia set can be determined by looking at the critical orbit, e.g. by the method used in Hubbard trees.
For the function f(Z) = Z' = Z2 + C, the Mandelbrot set can be defined as the set of all values C that produce a connected Julia set. This is also true for the Multibrot sets defined by f(Z) = Z' = Zp + C for integer P>2.
When viewing the Mandelbrot Set, one often encounters shapes that are reminiscent of Julia sets. If you are viewing an area of the Mandelbrot Set near a point P, then the most obvious features of the filaments in the area will resemble the Julia Set for C=P. However, you can find features of all of the Fatou Dusts near any point in the Mandelbrot Set. To find them you need to look near the island mu-molecules (see embedded Julia sets).
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013. Mu-ency index