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

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

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

:$f^n = \underbrace_n .$

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

Ergodic theory
Limit sets
Theorems in dynamical systems
Complex dynamics

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