No-wandering-domain theorem

No-wandering-domain theorem (Redirected from No wandering domain theorem) Aller à la navigation Aller à la recherche En mathématiques, 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. Plus précisément, for every component U in the Fatou set of f, la séquence {style d'affichage U,F(tu),F(F(tu)),des points ,f ^{n}(tu),des points } will eventually become periodic. Ici, f n denotes the n-fold iteration of f, C'est, {style d'affichage f^{n}=underbrace {fcirc fcirc cdots circ f} _{n}.} This image illustrates the dynamics of {style d'affichage 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; par exemple, the transcendental map {style d'affichage f(z)=z+2pi sin(z)} has wandering domains. Cependant, 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, ISBN 0-387-97942-5 MR1230383 Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. je. Solution of the Fatou-Julia problem on wandering domains, Annales de Mathématiques 122 (1985), non. 3, 401–18. MR0819553 S. Zakeri, Sullivan's proof of Fatou's no wandering domain conjecture This chaos theory-related article is a stub. Vous pouvez aider Wikipédia en l'agrandissant.

Catégories: Ergodic theoryLimit setsTheorems in dynamical systemsComplex dynamicsChaos theory stubs

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