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

${\displaystyle U,f(U),f(f(U)),\dots ,f^{n}(U),\dots }$

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

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

This image illustrates the dynamics of ${\displaystyle f(z)=z+2\pi \sin(z)}$; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.

The theorem does not hold for arbitrary maps; for example, the transcendental map ${\displaystyle f(z)=z+2\pi \sin(z)}$ 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 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