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Certain Julia sets include smooth components

In: Fractals and Chaos

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  • Benoit B. Mandelbrot

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

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.

Suggested Citation

  • Benoit B. Mandelbrot, 2004. "Certain Julia sets include smooth components," Springer Books, in: Fractals and Chaos, chapter 0, pages 114-116, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-4017-2_9
    DOI: 10.1007/978-1-4757-4017-2_9
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