> Periodic Orbits, External Rays, and the Mandelbrot Set: An Expository Account
> John Milnor
> Abstract: A presentation of some fundamental results from the Douady-Hubbard theory of the Mandelbrot set, based on the idea of "orbit portrait": the pattern of external rays landing on a periodic orbit for a quadratic polynomial map.

Example: ray pair (t--, t+) = (.(001010), (010001))

The lower ray t- has 6 images under angle doubling, so r p = 6. This collection includes t+.

The forward orbit of {t-, t+} \subset A_1 gives 3 distinct sets, consistent with p = 3. This makes r = 2, and thus this is a satellite component with v = r > 1, and v = 2 means A_1 = {t-,t+}.

Numbering the angles in A_1 = {t-,t+} as 0 <= t- = t^(1) < t^(2) = t+ < 1, one sees that 2^p t^(j) = t^(j+k) = t^(j+1), working in R/Z for ray angles and (mod v) for superscripts. Thus the combinatorial rotation number is k/v = 1/2,

So the ray pair (t-,t+) lands on the root of the 1/2-satellite of a period 3 component.

The nearest surrounding ray pair of period 3 is (.(001), .(010)). Under angle doubling, there are a total of v' p' = 3 rays in the orbit portrait, and v' >= 2 as the rays land in (at least) pairs, so v' = 3 and p' = 1 by basic properties of integer divisibility. So A'_1 = { .(001), .(010), .(100) }, from which ordering the combinatorial rotation number is seen to be 1/3.

So (.(001010), (010001) lands on the root of the 1/2-satellite of the 1/3-satellite of the period 1 cardioid.

In fact I constructed the initial ray pair by tuning: S(1/2) * S(1/3), so the result is verified.

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Example: ray pair (.(01110), .(10001)).

Under angle doubling there are 10 rays total, so v = 2 and p = 5. The lower ray has period 5 under angle doubling, so r p = 5, so r = 1, so this is a primitive component of period 5. Consider the surrounding lower period ray pairs (wakes) that separate this component from the parabolic root at c = 1/4:

Period 4: (.(0110), .(1001)) which is the satellite component S(1/2) * S(1/2) (note that .(0111), .(1000) is inside the period 5 wake, not outside)
Period 3: (.(011), (.100)).
Period 2: (.(01), .(10)) = S(1/2)
Period 1: (.(0), .(1))

Using the notation of Dierk Schleicher's "angled internal addresses", this component is called
1 ->_{1/2} 2 ->_{1/2} ->_{1/2} 3 ->_{1/2} 5

Example: single ray .(001001010) = t

Images under angle doubling:
.(000100101)
.(001001010) = t
.(001010001)
.(010001001)
.(010010100)
.(010100010)
.(100010010)
.(100101000)
.(101000100)
The smallest ("characteristic") sector between neighbouring rays is at (t-,t+) = ((001001010), .(001010001)), and t = t- so it is the lower ray of a satellite component of ray period r p = 9. (afaict, If the characteristic sector didn't include t as an endpoint it would be a primitive component and the other ray would need to be found another way)

t- and t+ land together, and t+ = 2^3 t- (mod 1), so p = 3 and k = 1. The valence v = 3 because r p = 9 and v = r > 1 for a satellite component. The other ray in A_1 is 2^3 t+ (mod 1). The combinatorial rotation number is k/v = 1/3. The component's angled internal address is 1 ->_{1/3} 3 ->_{1/3} 9

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