*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 *z*_{o}=0 is attracted to a cycle of a period *n*.
For the "quadratic" parametrization
*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*

*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 *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 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 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 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

Contents
Previous: Introduction
Next: The primary Bulbs counting

*updated* 8 Sep 2013