No-wandering-domain theorem

In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985.
The theorem states that a rational map "f" : Ĉ → Ĉ with deg("f") ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere. More precisely, for every component "U" in the Fatou set of "f", the sequence
will eventually become periodic. Here, "f" "n" denotes the "n"-fold iteration of "f", that is,
The theorem does not hold for arbitrary maps; for example, the transcendental map formula_3 has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.