No-wandering-domain theorem explained

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),} ...

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

f ^{n} = \underbrace{f \circ f\circ … \circ f} _{n} . 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,
• 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.
• S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture