This article is from the Fractal FAQ, by Ermel Stepp firstname.lastname@example.org 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 the map.
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 polynomial functions.