page_title: No-wandering-domain theorem
hierarchical_section_title: No-wandering-domain theorem
caption_attribution_description: English: This is a picture of the dynamical plane for the transcendental entire function f ( z ) = z + 2 &#x03C0; sin &#x2061; ( z ) {\displaystyle f(z)=z+2\pi \sin(z)} . The Fatou set, being a union of wandering domains, is shown in white. The Julia set is drawn in shades of grey, motivated by Arnaud Chéritat's similar image for maps in the sine family. (Darker pixels spend more time in the upper half-plane before mapping near infinity.) The image was created using a general-purpose heuristic for determining whether a pixel intersects the Julia set, followed by an ad-hoc method for the colouring of the Julia set. (Some artifacts are visible in the colouring of the Julia set; this is due to the fact that only one point of each pixel is iterated. It should be principle, in principle, to reduce these artifacts by using a similar heuristic as the one applied in the first part of the process.)
context_page_description: 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 ≥ 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
  will eventually become periodic. Here, f ⁿ denotes the n-fold iteration of f, that is,
  The theorem does not hold for arbitrary maps; for example, the transcendental map 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.
context_section_description: 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
  will eventually become periodic.  Here, f ⁿ denotes the n-fold iteration of f, that is,
  The theorem does not hold for arbitrary maps; for example, the transcendental map  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.
caption: This image illustrates the dynamics of 
    
      
        
          f
          (
          z
          )
          =
          z
          +
          2
          π
          sin
          ⁡
          (
          z
          )
        
      
      {\displaystyle f(z)=z+2\pi \sin(z)}
    
  ; 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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