Certain Julia sets include smooth components

Benoit B. Mandelbrot
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Benoit B. Mandelbrot: Yale University, Mathematics Department

Chapter C9 in Fractals and Chaos, 2004, pp 114-116 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-1-4757-4017-2_9

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