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)),...,fn(U),...

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

fn=\underbrace{f\circf\circ\circf}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.

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