Advertisement

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

  • Benoit B. Mandelbrot
Chapter

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.

Keywords

Double Point Jordan Curve Golden Ratio Continue Fraction Expansion Fibonacci Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations