Rotation Numbers and Internal angles of
the Mandelbrot bulbs
Robert L. Devaney
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.
"The Mandelbrot cactus" ("square" parametrisation).
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 "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)
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).
 Robert L. Devaney The Fractal Geometry of the Mandelbrot Set II.
How to Count and How to Add:
3 Periods of the Bulbs
Next: The primary Bulbs counting
updated 12 February 2000