Julia gave a precise description about a function J(f), in which z is a complex number, for which the n-th element of sequence f^n(z) stays equal, while n is growing to infinity.
There is an infinite number Julia Sets wich are Subsets of the complex plane C. As you do calculating a Mandelbrot Set, you pick a point in C.
Z1 = Z0 + Z02
Z2 = Z1 + Z02
Z3 = Z2 + Z02
. . .
If the sequence Z0, Z1, Z2, Z3, ... remains within a distance of 2 of the origin forever, then the point Z0 is said to be in the Julia set. If the sequence diverges from the origin, then the point is not in the set. How fast the sequence diverges can be translated into a color.
If the point you pick is in the Mandelbrot Set, the resulting Julia Set is connected.