Rotation Numbers and Internal angles of
the Mandelbrot bulbs

Robert L. Devaney


"The Mandelbrot cactus" ("square" parametrisation).
The Mandelbrot set consists of many small decorations or bulbs (or limbs or atoms) [1]. 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 "square" parametrisation 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 → 180o, 1/3 → 120o and 1/4 → 90o)

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 counterclockwise direction at each iteration. Another way to say this is the cycle rotates by a m/ n revolution in the counterclockwise 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 counterclockwise 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 counterclockwise direction from the main spoke ("C" parametrisation here).
The rule follows from similarity of this antenna near Mizurevich point m and corresponding J-set (see below).

[1] Robert L. Devaney The Fractal Geometry of the Mandelbrot Set II.
How to Count and How to Add: 3 Periods of the Bulbs


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updated 12 February 2000