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

In: Fractals and Chaos

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Listed:
  • Benoit B. Mandelbrot

    (Yale University, Mathematics Department
    IBM T.J. Watson Research Center)

Abstract

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.

Suggested Citation

  • Benoit B. Mandelbrot, 2004. "Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs," Springer Books, in: Fractals and Chaos, chapter 0, pages 117-124, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-4017-2_10
    DOI: 10.1007/978-1-4757-4017-2_10
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