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):

• period of main pseudocardioid
• localization
  • internal adress
  • wake
  • center of main pseudocardioid ( nucleus)
  • viewport: rectangle part (window) of parameter plane used to show the island. It is usually descibed by radius and center
  • angles of external rays that land on cusp of the pseudocardiod
• orientation
• size
  • size of main pseudocardioid
• shape
  • distortion

Usually more then one measure can be used:

• biggest island of the wake
• biggest island with period

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

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 :

-1.76733 +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
• external angles (3/7,4/7) which in binary is (.(011),.(100))

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. 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:

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.

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 18

Period 18 island with angled internal address

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

whose:

• upper external angle is .(010101010101100101)
• 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

period 44

Plane parameters :

-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 :

  -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

Island with 116 period

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)

Angled internal address :

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

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) :

• start with center of period 1 ( c=0)
• 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

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

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

period 134

Period 134 island

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:

• 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:

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

The angled internal address of Old Wood Dish starts:

$11/2216/17331/2341/3691/2701/31411/21421/32851/2286...$

and the pattern can be extended indefinitely by

$...1/3(p-1)1/2p1/3(2p+1)1/2(2p+2)...$

See also

• How to compute external angles of principal Misiurewicz poin of wake p/q ?
• fractalforums.com : the-mandelbrot-polynomial-roots-challenge
• fractalforums.org: the-mandelbrot-set-root-challenge-10-years-later
• distortion of the islands ( mini Mandelbrot sets )