Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs

  • Benoit B. Mandelbrot


Within the M-set of the map z → λz(1-z), consider a sequence of points λm having a limit point λ. Denote the corresponding F* -sets by ℱ*(λm) and ℱ*(λ). In general, lim ℱ*(λm) = ℱ*(lim λm), but there is a very important exception. In some cases, the sets ℱ*(λm) do not converge to either a curve or a dust, but converge to a domain of the A -plane, part of which is called the Siegel disc l while the rest is made of the preimages of ℒ. In such cases, ℱ*(lim λm) is not the set lim ℱ*λm but only that set’s boundary. The intuitive meaning of this behavior is discussed and illustrated in terms of the so-called Peano curves, and a mathematical question is raised concerning the nonrational and non-Siegel λ.


Double Point Jordan Curve Golden Ratio Continue Fraction Expansion Fibonacci Sequence 
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Copyright information

© Benoit B. Mandelbrot 2004

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
    • 2
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA

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