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6.4: Embedded Julia midgets

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    M and J-sets similarity. Embedded Julia midgets

    Tan Lei's theorem says, that the M-set at a preperiodic point (e.g. at the band merging point M = -1.543689 in the second picture) and the Julia set J(M) at z = M are similar and have the same scaling constant.

    6.4.1.png
    Figure \(\PageIndex{1}\)
    6.4.2.png
    Figure \(\PageIndex{2}\)
    6.4.3.png
    Figure \(\PageIndex{3}\)

    In the vicinity of the periodic point P = -1.75488 (in the center of the period-3 midget) the Mandelbrot set is similar again to the Julia set J(P). As since fco3 is renormalizable for c = P , we see the J(0) set at z = P and its shrinking copies (instead of sequence of tiny midgets in the M-set).

    At last this image corresponds to A = -1.7465234 at the cusp of the period-3 midget. The point A lays from outside of tiny cardioid, therefore we see small copies of the cauliflower Cantor dust J(0.35) set.

    6.4.4.png
    Figure \(\PageIndex{4}\)
    6.4.5.jpg
    Figure \(\PageIndex{5}\)
    6.4.6.jpg
    Figure \(\PageIndex{6}\)

    To make the last step we look at the M-set near A . As since at small scale the M-set is similar to corresponding J(A) set, we see embedded J(0.35) midget in the Mandelbrot set (in the right applet). Note however, that this "cauliflower" is replaced by the M-midget in the left window. Every embedded cauliflower has a tiny M-midget in its center.

    For more details see Robert P. Munafo's Embedded Julia Set.

     

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