Definition of Julia Set

Definition 2.4.1. Let f: X  X is an analytic self-map of a Riemann surface X. We will assume that X is the Riemann sphere, the complex plane, or the once-punctured complex plane, as the other cases do not give rise to interesting dynamics. (Such maps are completely classified.)
It will consider f as a discrete dynamical system on the phase space X, so we are interested in the behavior of the iterates fn of f (that is, the n-fold compositions of f with itself).
The Fatou set of f consists of all points z  X such that the family of iterates , forms a normal family in the sense of Montel when restricted to some open neighborhood of z. The Julia set of f is the complement of the Fatou set in X.