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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 ≥ 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 denotes the n-fold iteration of f, that is,
The theorem does not hold for arbitrary maps; for example, the transcendental map f = z + 2 π sin {\displaystyle f=z+2\pi \sin} 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.