This article is from the Fractal FAQ, by Ermel Stepp firstname.lastname@example.org with numerous contributions by others.
The Julia set can be computed by iteration similar to the Mandelbrot
computation. The only difference is that the c value is fixed and the
initial z value varies.
Alternatively, points on the boundary of the Julia set can be computed
quickly by using inverse iterations. This technique is particularly
useful when the Julia set is a Cantor Set. In inverse iteration, the
equation z1 = z0^2+c is reversed to give an equation for
z0: z0 = +- sqrt(z1-c). By applying this equation repeatedly, the
resulting points quickly converge to the Julia set boundary. (At each
step, either the postive or negative root is randomly selected.) This
is a nonlinear iterated function system. In pseudocode: z = 1 (or any
if (random number < .5) then
z = sqrt(z-c)