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

$U,f(U),f(f(U)),\dots,f^n(U), \dots$

will eventually become periodic. Here, f ⁿ denotes the n-fold iteration of f, that is,

$f^n = \underbrace{f \circ f\circ \cdots \circ f}_n .$

The theorem does not hold for arbitrary maps; for example, the transcendental map f(z) = z + sin(2πz) has wandering domains.

References

• Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993, ISBN 0-387-97942-5 MR1230383
• 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. MR0819553
• S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture

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