How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?

Examples

period 3 island

Wakes near the period 3 island in the Mandelbrot set. Boundary of the Mandelbrot set rendered with distance estimation (exterior and interior). Labelled with periods (blue), internal angles and rays (green) and external angles and rays (red).

find angles of some child bulbs of period 3 component ( island) on main antenna with external rays (3/7,4/7)

Plane description :^[1]

-1.76733 +0.00002 i @ 0.05

One can check it using program Mandel by Wolf Jung :

The angle  3/7  or  p011 has  preperiod = 0  and  period = 3.
The conjugate angle is  4/7  or  p100 .
The kneading sequence is  AB*  and the internal address is  1-2-3 .
The corresponding parameter rays are landing at the root of a primitive component of period 3.

Period 5 islands

on the main antenna

Wakes along the main antenna in the Mandelbrot set. Boundary of the Mandelbrot set rendered with distance estimation (exterior and interior). Labelled with periods (blue), internal addresses (green) and external angles and rays (red).

There are 3 period 5 componenets on the main antenna ( checked with program Mandel by Wolf Jung ) :

The angle  13/31  or  p01101 has  preperiod = 0  and  period = 5.
The conjugate angle is  18/31  or  p10010 .
The kneading sequence is  ABAA*  and the internal address is  1-2-4-5 .
The corresponding parameter rays are landing at the root of a primitive component of period 5.

The angle  14/31  or  p01110 has  preperiod = 0  and  period = 5.
The conjugate angle is  17/31  or  p10001 .
The kneading sequence is  ABBA*  and the internal address is  1-2-3-5 .
The corresponding parameter rays are landing at the root of a primitive component of period 5.

The angle  15/31  or  p01111 has  preperiod = 0  and  period = 5.
The conjugate angle is  16/31  or  p10000 .
The kneading sequence is  ABBB*  and the internal address is  1-2-3-4-5 .
The corresponding parameter rays are landing at the root of a primitive component of period 5.

1-3-4-5

Angled internal address in the form used by Claude Heiland-Allen:^[2]

1 1/2 2 1/2 3 1/2 4 1/2 5

or in the other form :

 $1{\xrightarrow {Sharkovsky}}3{\xrightarrow {1/2}}\cdots {\xrightarrow {}}4{\xrightarrow {1/2}}\cdots {\xrightarrow {}}5$

Where

$1{\xrightarrow {Sharkovsky}}3$ denotes Sharkovsky ordering which describes what is going on between period 1 and 3 on the real axis. It's first part is period doubling scenario from period 1 : $1{\xrightarrow {1/2}}\cdots$ denotes $1{\xrightarrow {1/2}}2{\xrightarrow {1/2}}4{\xrightarrow {1/2}}8{\xrightarrow {1/2}}16{\xrightarrow {1/2}}32{\xrightarrow {1/2}}1*2^{n}....$
$p{\xrightarrow {1/2}}\cdots$ denotes period p component and infinite number of boundary crossing along 1/2 internal rays, for example $3{\xrightarrow {1/2}}\cdots$ denotes $3{\xrightarrow {1/2}}6{\xrightarrow {1/2}}12{\xrightarrow {1/2}}24{\xrightarrow {1/2}}48{\xrightarrow {1/2}}3*2^{n}....$

So going from period 1 to period 5 on the main antenna means infinite number of boundary crossing ! It is to much so one has to start from main component of period 5 island.

External angles of this componnet can be computed by other algorithms.^[3]

1/5

Choose

  $1{\xrightarrow {Sharkovsky}}3{\xrightarrow {1/2}}\cdots {\xrightarrow {}}4{\xrightarrow {1/2}}\cdots {\xrightarrow {}}5{\xrightarrow {1/5}}25$

First compute external angles for r/s wake :

 $\theta _{-}(r/s)=\theta _{-}(1/5)=0.(00001)$ 
 $\theta _{+}(r/s)=\theta _{+}(1/5)=0.(00010)$

and root of the island ( using program Mandel ) :

The angle  13/31  or  p01101
has  preperiod = 0  and  period = 5.
The conjugate angle is  18/31  or  p10010 .
The kneading sequence is  ABAA*  and
the internal address is  1-2-4-5 .
The corresponding parameter rays are landing
at the root of a primitive component of period 5.

 $\theta _{-}(island)=0.({\color {Blue}01101})$ 
 $\theta _{+}(island)=0.({\color {Red}10010})$

then in $\theta (r/s)$ replace :

digit 0 by block of length q from $\theta _{-}(island)$
digit 1 by block of length q from $\theta _{+}(island)$

Result is :

 $\theta _{-}(island,r/s)=\theta _{-}(5,1/5)=0.({\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Red}10010})$ 
 $\theta _{+}(island,r/s)=\theta _{+}(5,1/5)=0.({\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Blue}01101}\ {\color {Red}10010}\ {\color {Blue}01101})$

theta_minus = 0.(0110101101011010110110010)
theta_plus  = 0.(0110101101011011001001101)

One can check it using program Mandel by Wolf Jung :

The angle  14071218/33554431  or  p0110101101011010110110010
has  preperiod = 0  and  period = 25.
The conjugate angle is  14071373/33554431  or  p0110101101011011001001101 .
The kneading sequence is  ABAABABAABABAABABAABABAA*  and
the internal address is  1-2-4-5-25 .
The corresponding parameter rays are landing
at the root of a satellite component of period 25.
It is bifurcating from period 5.
Do you want to draw the rays and to shift c
to the corresponding center?

period 9 island

Part of parameter plane with Minimandelbrot sets for periods 1, 3, 9, 27, 81, 243. Also external arays are seen.

the period 9 island in the antenna of the period 3 island

Check with Mandel:

The angle  228/511  or  p011100100 has  preperiod = 0  and  period = 9.
The conjugate angle is  283/511  or  p100011011 .
The kneading sequence is  ABBABAAB*  and the internal address is  1-2-3-6-9 .
The corresponding parameter rays are landing at the root of a primitive component of period 9.

period 32

+0.2925755 -0.0149977i @ +0.0005 ^[4]

period 44

Plane parameters :^[5]

-0.63413421522307309166332840960 + 0.68661141963581069380394003021 i @ 3.35e-24

and external rays :

.(01001111100100100100011101010110011001100011)
.(01001111100100100100011101010110011001100100)

One can check it with program Mandel by Wolf Jung :

The angle  5468105041507/17592186044415  or  p01001111100100100100011101010110011001100011
has  preperiod = 0  and  period = 44.
The conjugate angle is  5468105041508/17592186044415  or  p01001111100100100100011101010110011001100100 .
The kneading sequence is  AAAABBBBABAABAABAABAABBBABABABAAABAAABABAAB*  and
the internal address is  1-5-6-7-8-10-13-16-19-22-23-24-26-28-30-34-38-40-43-44 .
The corresponding parameter rays are landing
at the root of a primitive component of period 44.

period 52

Plane parameters :^[6]

  -0.22817920780250860271129306628202459167994 +   1.11515676722969926888221122588497247465766 i @ 2.22e-41

and external rays :

.(0011111111101010101010101011111111101010101010101011)
.(0011111111101010101010101011111111101010101010101100)

One can check it with program Mandel by Wolf Jung :

The angle  1124433913621163/4503599627370495  or  p0011111111101010101010101011111111101010101010101011
has  preperiod = 0  and  period = 52.
The conjugate angle is  1124433913621164/4503599627370495  or  p0011111111101010101010101011111111101010101010101100 .
The kneading sequence is  AABBBBBBBBBABABABABABABABABBBBBBBBBABABABABABABABAB*  and
the internal address is  1-3-4-5-6-7-8-9-10-11-13-15-17-19-21-23-25-27-28-29-30-31-32-33-34-35-37-39-41-43-45-47-49-51-52 .
The corresponding parameter rays are landing
at the root of a primitive component of period 52.

period 134

Period 134 island

References

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[1] R2F(1/2B1)S by Robert P. Munafo, 2008 Feb 28.

[2] Patterns of periods in the Mandelbrot set by Claude Heiland-Allen

[3] Parameter rays of root points of period p components

[4] R2.C(0) by Robert P. Munafo, 2012 Apr 16.

[5] Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen

[6] Navigating by spokes in the Mandelbrot set by Claude Heiland-Allen