no-wandering-domain theorem

{{Short description|Mathematical 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

:$U,f(U),f(f(U)),\backslash dots,f^n(U),\; \backslash dots$

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

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

File:Wandering domains for the entire function f(z)=z+2πsin(z).png

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