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1.1: Rotation Numbers and Internal angles of the Mandelbrot bulbs

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    101023
    • Robert L. Devaney

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    "The Mandelbrot cactus" ("quadratic" parametrization).

    The Mandelbrot set consists of many small decorations or bulbs (or limbs or atoms). A decoration directly attached to the main cardioid in M is called a primary bulb. This bulb in turn has infinitely many smaller bulbs attached. It is known that if c lies in the interior of a bulb, then the orbit of zo=0 is attracted to a cycle of a period n. For the "quadratic" parametrization c = 1/4 - a2 

    zn+1 = zn2 + 1/4 - a2

    the main cardioid of the M-set turns into a circle with radius r = 1/2. A primary bulb attaches to the main circle at an internal angle   

    φ = 2 π m/n

    where m/n is rotation number (e.g. 1/2 → 180 o, 1/3 → 120 o and 1/4 → 90 o).

    1.1.1.jpg
    Figure \(\PageIndex{1}\): -0.7500  0.0000  1.5100   440ms
    1.1.2.png1.1.3.jpg
    Figure \(\PageIndex{2}\): 0.0500  0.0000  2.6700   72ms
    1.1.3.jpg
    Figure \(\PageIndex{3}\): 0.0500  0.0000  2.9000   69ms

    1. One can count rotation number of a bulb by its periodic orbit star on dynamical plane. An attracting period n cycle z1 → z2 →...→ zn → z1 hops among zi as fc is iterated. If we observe this motion, the cycle jumps exactly m points in the counter-clockwise direction at each iteration. Another way to say this is the cycle rotates by a m/ n revolution in the counter-clockwise direction under iteration.
    2. The Jc-set contains infinitely many "junction points" at which n distinct black regions in J-set are attached, because c- value lies in a primary period n (3 or 5 for these images) bulb in the M-set. And the smallest black region is located m revolutions in the counter-clockwise direction from the largest central region.
    3. The number of spokes in the largest antenna attached to a primary decoration of M-set is equivalent to the period of that decoration. And the shortest spoke is located m revolutions in the counter-clockwise direction from the main spoke ("C" parametrization here).

    1.1.4.png
    Figure \(\PageIndex{4}\): -0.1175  0.9000  0.5000   246ms
    1.1.5.png
    Figure \(\PageIndex{5}\): -0.5400  0.6100  0.2000   370ms

    The rule follows from similarity of an antenna near a Mizurevich point and corresponding J-set (see The M and J-sets similarity).

    1.1.6.jpg
    Figure \(\PageIndex{6}\): -0.0500  0.0000  2.8000   315ms
    1.1.7.png
    Figure \(\PageIndex{7}\): -0.0500  0.0000  2.8000   341ms

     

    This page titled 1.1: Rotation Numbers and Internal angles of the Mandelbrot bulbs was last modified on Thu, 21 Apr 2022 21:30:36 GMT and is shared under a Public Domain license and was authored, remixed, and/or curated by Robert L. Devaney via source content that was edited to the style and standards of the LibreTexts platform.