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Certain Julia sets include smooth components

  • Benoit B. Mandelbrot
Chapter

Abstract

The Julia set F* of the map z → ỹ(z) = z2-μ may be the boundary of an atom, of a molecule, or of a “devil’s polymer” in the z-plane. Denote the boundary of one of the atoms of F* by H. When μ ≠ 0 is the nucleus of a cardioid-shaped atom of the M-set, it is conjectured that the fractal dimension D of H is 1. Thus, H may be a be a rectifiable curve (of well defined length) or perhaps only a borderline fractal curve (of logarithmically diverging length). This paper comments on a clearer version of Figure 5 of M19831{C5} and develops a remark made there, but not very explicitly.

Keywords

Fractal Dimension Jordan Curve Algebraic Curf Real Interval Simple Loop 
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.

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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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