No-wandering-domain theorem

No-wandering-domain theorem (Redirected from No wandering domain theorem) Zur Navigation springen Zur Suche springen In der Mathematik, 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. Etwas präziser, for every component U in the Fatou set of f, die Sequenz {Anzeigestil U,f(U),f(f(U)),Punkte ,f^{n}(U),Punkte } will eventually become periodic. Hier, f n denotes the n-fold iteration of f, das ist, {Anzeigestil f^{n}=underbrace {fcirc fcirc cdots circ f} _{n}.} This image illustrates the dynamics of {Anzeigestil f(z)=z+2pi sin(z)} ; the Fatou set (consisting entirely of wandering domains) is shown in white, while the Julia set is shown in tones of gray.

The theorem does not hold for arbitrary maps; zum Beispiel, the transcendental map {Anzeigestil f(z)=z+2pi sin(z)} has wandering domains. Jedoch, 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, Universitätstext: Tracts in Mathematics, Springer-Verlag, New York, 1993, ISBN 0-387-97942-5 MR1230383 Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. ich. Solution of the Fatou-Julia problem on wandering domains, Annalen der Mathematik 122 (1985), nein. 3, 401–18. MR0819553 S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture This chaos theory-related article is a stub. Sie können Wikipedia helfen, indem Sie es erweitern.

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