**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 *z*_{0}=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} - a^{2}

*z*_{n+1} = z_{n}^{2} +
^{1}/_{4} - a^{2}

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} -> 180^{o},
^{1}/_{3} -> 120^{o} and
^{1}/_{4} -> 90^{o})

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

1. One can count rotation number of a bulb by its periodic orbit star.
An attracting period *n* cycle *z*_{1} -> z_{2}
->...-> z_{n} -> z_{1} hops among *z*_{i} as
*f*_{c} 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 J_{c}-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

Contents
Previous: Introduction
Next: The primary Bulbs counting

*updated* 12 February 2000