Robert L. Devaney

Rotation Numbers and Internal angles of the Mandelbrot bulbs

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

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

[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 8 Sep 2013