This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
The Julia set of any rational map of degree greater than one is
perfect (hence in particular uncountable and nonempty), completely
invariant, equal to the Julia set of any iterate of the function, and
also is the boundary of the basin of attraction of every attractor for
Julia set references:
1. A. F. Beardon, "Iteration of Rational Functions : Complex Analytic
Dynamical Systems", Springer-Verlag, New York, 1991.
2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere,
"Bull. of the Amer. Math. Soc" 11, 1 (July 1984), pp. 85-141.
This article is a detailed discussion of the mathematics of iterated
complex functions. It covers most things about Julia sets of rational