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| In [[mathematics]], the '''no-wandering-domain theorem''' is a result on [[dynamical system]]s, proven by [[Dennis Sullivan]] in 1985.
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| The theorem states that a [[rational function|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 [[Connected space|component]] ''U'' in the [[Fatou set]] of ''f'', the sequence
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| :<math>U,f(U),f(f(U)),\dots,f^n(U), \dots</math>
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| will eventually become periodic. Here, ''f''<sup> ''n''</sup> denotes the [[function iteration|''n''-fold iteration]] of ''f'', that is,
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| :<math>f^n = \underbrace{f \circ f\circ \cdots \circ f}_n .</math> | |
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| [[File:Wandering_domains_for_the_entire_function_f%28z%29%3Dz%2B2πsin%28z%29.png|thumb|alt=An image of the dynamical plane for <math>f(z)=z+2\pi\sin(z)</math>.|This image illustrates the dynamics of <math>f(z)=z+2\pi\sin(z)</math>; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.]]
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| The theorem does not hold for arbitrary maps; for example, the [[Transcendental function|transcendental map]] <math>f(z)=z+2\pi\sin(z)</math> 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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| ==References==
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| * [[Lennart Carleson]] and Theodore W. Gamelin, ''Complex Dynamics'', Universitext: Tracts in Mathematics, [[Springer-Verlag]], New York, 1993, ISBN 0-387-97942-5 {{MathSciNet| id=1230383}}
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| * 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. {{MathSciNet| id=0819553}}
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| * S. Zakeri, ''[http://www.math.qc.edu/~zakeri/notes/wander.pdf Sullivan's proof of Fatou's no wandering domain conjecture]''
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| {{mathapplied-stub}}
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| [[Category:Ergodic theory]]
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| [[Category:Limit sets]]
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| [[Category:Theorems in dynamical systems]]
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| [[Category:Complex dynamics]]
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. To perform domino is something I really enjoy doing. She functions as a journey agent but quickly she'll be on her own. My wife and I live in Mississippi but now I'm considering other choices.
my website - real psychics