No-wandering-domain theorem

In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985.

More precisely, for every component U in the Fatou set of f, the sequence will eventually become periodic.

Here, f n denotes the n-fold iteration of f, that is, The theorem does not hold for arbitrary maps; for example, the transcendental map

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.

This chaos theory-related article is a stub.

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