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
U , f ( U ) , f ( f ( U ) ) , … , f n ( U ) , … {\displaystyle U,f(U),f(f(U)),\dots ,f^{n}(U),\dots }
will eventually become periodic. Here, f n denotes the n-fold iteration of f, that is,
f n = f ∘ f ∘ ⋯ ∘ f ⏟ n . {\displaystyle f^{n}=\underbrace {f\circ f\circ \cdots \circ f} _{n}.}
The theorem does not hold for arbitrary maps; for example, the transcendental map f ( z ) = z + 2 π sin ( z ) {\displaystyle f(z)=z+2\pi \sin(z)} 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.
References Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993, ISBN 0-387-97942-5 MR 1230383 Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Annals of Mathematics 122 (1985), no. 3, 401–18. MR 0819553 S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture