Rotation Numbers and Internal angles of
the Mandelbrot bulbs

Robert L. Devaney

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 z0=0 is attracted to a cycle of a period n. It is a multiple of n for c inside the other smaller bulbs attached to the primary bulb.
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
    f = 2 p m/n
where m/n is rotation number (e.g. 1/2 -> 180o, 1/3 -> 120o and 1/4 -> 90o)

"The Mandelbrot cactus" ("square" parametrisation).

1. One can count rotation number of a bulb by its periodic orbit star. 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 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).

[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