# Fractals/Iterations in the complex plane/island t

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 ?

# How to describe island ?

Criteria for classifications ( measures):

Usually more then one measure can be used:

# Islands by period

const roots = [
[0, 0],
[-1.98542425305421, 0,'Needle Far Left'],
[-1.86078252220485, 0,'Needle Not So Far Left'],
[-1.6254137251233, 0,'Needle Near'],
[-1.25636793006818, -0.380320963472722, "Biggest Minibrot Lower Left"],
[-1.25636793006818 , 0.380320963472722, "Biggest Minibrot Upper Left"],
[-0.504340175446244 ,-0.562765761452982, "Bulb MainLeftLower"],
[-0.504340175446244 ,0.562765761452982, "Bulb MainLeftUpper"],
[-0.0442123577040706 ,-0.986580976280893,"Minibrot Lower Right"],
[-0.0442123577040706 , 0.986580976280893,"Minibrot Upper Right"],
[-0.198042099364254 ,-1.1002695372927,'#Deeper Minibrot Lower Left'],
[-0.198042099364254 , 1.1002695372927,'#Deeper Minibrot Upper Left'],
[0.379513588015924 ,-0.334932305597498,"Bulb MainRightLower"],
[0.379513588015924 ,+ 0.334932305597498,"Bulb MainRightUpper"],
[0.359259224758007 ,-0.642513737138542,"Minibrot MainRightLower Back"],
[0.359259224758007 , 0.642513737138542,"Minibrot MainRightUpper Back"]
]


## period 3 island

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.


The largest mini on the antenna has:

• internal adress 1 1/2 2 1/2 3[2]
• external angles (3/7,4/7) which in binary is (.(011),.(100))

## Period 5 islands

### on the main antenna

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. On the next root land 2 rays 16/33 i 17/33

#### 1-3-4-5

Angled internal address in the form used by Claude Heiland-Allen:[3]

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


or in the other form :

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


Where

• ${\displaystyle 1{\xrightarrow {Sharkovsky}}3}$  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 : ${\displaystyle 1{\xrightarrow {1/2}}\cdots }$  denotes ${\displaystyle 1{\xrightarrow {1/2}}2{\xrightarrow {1/2}}4{\xrightarrow {1/2}}8{\xrightarrow {1/2}}16{\xrightarrow {1/2}}32{\xrightarrow {1/2}}1*2^{n}....}$
• ${\displaystyle p{\xrightarrow {1/2}}\cdots }$  denotes period p component and infinite number of boundary crossing along 1/2 internal rays, for example ${\displaystyle 3{\xrightarrow {1/2}}\cdots }$  denotes ${\displaystyle 3{\xrightarrow {1/2}}6{\xrightarrow {1/2}}12{\xrightarrow {1/2}}24{\xrightarrow {1/2}}48{\xrightarrow {1/2}}3*2^{n}....}$

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.[4]

##### 1/5

Choose

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


First compute external angles for r/s wake :

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


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.


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


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

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

Result is :

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

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

• 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 18

Period 18 island with angled internal address

 ${\displaystyle 1\xrightarrow {1/2} 2\xrightarrow {1/8} 16\xrightarrow {1/2} ...\xrightarrow {} 18}$



whose:

• upper external angle is .(010101010101100101) [5]
• center ( nucleus) c = -0.814158841137593 +0.189802029306573 i

Info from progrm Mandel :

The angle  87397/262143  or  p010101010101100101 has  preperiod = 0  and  period = 18.
The conjugate angle is  87386/262143  or  p010101010101011010 .
The kneading sequence is  ABABABABABABABAAA*  and the internal address is  1-2-16-18 .
The corresponding parameter rays land at the root of a primitive component of period 18.


## period 16

• +0.2925755 -0.0149977i @ +0.0005 [6]

## period 44

Plane parameters :[7]

-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 49

• center c = -0.748427377115632 +0.067417674789180 i period = 49
• distorted
• in the wake of c = -0.747115035379558 +0.066741875885198 i period = 47

## period 52

Plane parameters :[8]

  -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.


render using MPFR ( double precision is not enough)

## period 61

• center c = -0.749007413067268 +0.053603465229520 i period = 61
• distorted
The 29/59-wake of the main cardioid is bounded by the parameter rays with the angles
192153584101141161/576460752303423487  or  p01010101010101010101010101010101010101010101010101010101001  and
192153584101141162/576460752303423487  or  p01010101010101010101010101010101010101010101010101010101010 .
Do you want to draw the rays and to shift c to the center of the satellite component?
c = -0.748168212862783  +0.053193574107985 i    period = 59


## period 116

It is inside 5/11 wake

size 1000 1000
view 53 -7.2398344555005190e-01 2.8671972540880530e-01 8.0481388661397700e-07
text 53 -7.2398348100841969e-01 2.8671974646855508e-01 116
ray_in 2000 .(01010101001101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001010101010)
ray_in 3000 .(01010101001101001010101010010101010100101010101001010101010010101010100101010101001010101010010101010100101010101001)


${\displaystyle 1\xrightarrow {5/11} 11\xrightarrow {1/2} 22\xrightarrow {1/2} 33\xrightarrow {1/2} 44\xrightarrow {1/2} 55\xrightarrow {1/2} 66\xrightarrow {1/2} 77\xrightarrow {1/2} 88\xrightarrow {1/2} 99\xrightarrow {1/2} 110\xrightarrow {1/2} 116}$

Above approach above address seems true but not practical.

Visual analysis gives full path inside Mandelbrot set ( more precisely inside main cardioid and 5/11-limb) :

• go along interna ray 5/11 to root ( bond)
• go to the period 11 center
• go along escape route 1/2 (thru period doubling cascade , Myrberg-Feigenbaum point and chotic part ) to principal Misiurewicz point of 5/11 wake: M_{11,1} = c = -0.724112682973574 +0.286456567676711 i [/li]
• turn into 3 branch
• go "straight" along the branch until center of period 116

${\displaystyle 1\xrightarrow {5/11} 11\xrightarrow {1/2} M_{11,1}\to ThirdBranch\to 116}$

There are ininite number of hyperbolic componnets inside branch, chaotic part and period doubling cascade so ther is no need to list them.

## period 275

• center of main pseudocardioid c = -1.985467748182376 +0.000003464322064 i period = 275
• distorted
• in the wake of c = -1.985424253054205 +0.000000000000000 i period = 5

## period 3104

Description:[9]

• Real number position: 0,25000102515011806826817597033225524583655
• Imaginary number position: 0,0000000016387052819136931666219461
• Zoom: 6,871947673*(10^10)
• bits = 38 , use mpfr type
• wake 1/3103

## Old Wood Dish

The atom periods of the center of Old Wood Dish are:[10]

1, 2, 34, 70, 142, 286, 574, 862, 1438, 2878, 5758


The angled internal address of Old Wood Dish starts:

${\displaystyle 11/2216/17331/2341/3691/2701/31411/21421/32851/2286...}$

and the pattern can be extended indefinitely by

${\displaystyle ...1/3(p-1)1/2p1/3(2p+1)1/2(2p+2)...}$