Fractals/Iterations in the complex plane/subwake

< Fractals < Iterations in the complex plane

How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is accesible from main cardioid ( M0) by a finite number of boundary crossing ?^[1]

Algorithm = Douady tuning

"The r/s internal ray in $M_{p/q}$ is the landing point of external rays $\theta _{\pm }(p/q,r/s)$ obtained from $\theta _{\pm }(r/s)$ by replacing:

the digit 0 by repeating block ( of length q) from $\theta _{-}(p/q)$
the digit 1 by repeating block ( of length q) from $\theta _{+}(p/q)$ "

"By repeating the same process ( which is known as 'tuning') we can compute the arguments of external rays landing on the boundary of any component which is accesible from $M_{0}$ by a finite number of boundary crossing." ( Shaun Bullett )

Douady tuning ^[2]

Examples

Wakes near the period 1 continent 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).

2 angles

(1/3,1/2)

First compute external angles for p/q and r/s wakes :

 $\theta _{-}(r/s)=\theta _{-}(1/2)=0.(01)$ 
 $\theta _{+}(r/s)=\theta _{+}(1/2)=0.(10)$

 $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$ 
 $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

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

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

Result is :

 $\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Red}010})$ 
 $\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Red}010}\ {\color {Blue}001})$

One can check it using program Mandel by Wolf Jung :

The angle  10/63  or  p001010
has  preperiod = 0  and  period = 6.
The conjugate angle is  17/63  or  p010001 .
The kneading sequence is  AABAA*  and
the internal address is  1-3-6 .
The corresponding parameter rays are landing
at the root of a satellite component of period 6.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,1/3)

First compute external angles for p/q and r/s wakes ( here p/q=r/s) :

 $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$ 
 $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

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

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

Result is :

 $\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$ 
 $\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$

One can check it using program Mandel by Wolf Jung :

The angle  74/511  or  p001001010
has  preperiod = 0  and  period = 9.
The conjugate angle is  81/511  or  p001010001 .
The kneading sequence is  AABAABAA*  and
the internal address is  1-3-9 .
The corresponding parameter rays are landing
at the root of a satellite component of period 9.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,1/4)

First compute external angles for p/q and r/s wakes :

 $\theta _{-}(r/s)=\theta _{-}(1/4)=0.(0001)$ 
 $\theta _{+}(r/s)=\theta _{+}(1/4)=0.(0010)$

 $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$ 
 $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

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

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

Result is :

 $\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/4)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$ 
 $\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/4)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$

One can check it using program Mandel by Wolf Jung :

The angle  586/4095  or  p001001001010
has  preperiod = 0  and  period = 12.
The conjugate angle is  593/4095  or  p001001010001 .
The kneading sequence is  AABAABAABAA*  and
the internal address is  1-3-12 .
The corresponding parameter rays are landing
at the root of a satellite component of period 12.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,3/4)

First compute external angles for p/q and r/s wakes :

 $\theta _{-}(r/s)=\theta _{-}(3/4)=0.(1101)$ 
 $\theta _{+}(r/s)=\theta _{+}(3/4)=0.(1110)$

 $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$ 
 $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

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

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

Result is :

 $\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Red}010}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Red}010})$ 
 $\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Red}010}\ {\color {Red}010}\ {\color {Red}010}\ {\color {Blue}001})$

One can check it in program Mandel by Wolf Jung :

The angle  1162/4095  or  p010010001010
has  preperiod = 0  and  period = 12.
The conjugate angle is  1169/4095  or  p010010010001 .
The kneading sequence is  AABAABAABAA*  and
the internal address is  1-3-12 .
The corresponding parameter rays are landing
at the root of a satellite component of period 12.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/4, 1/5)

Input is :

 $(p/q,r/s)=(1/4,1/5)$

First compute external angles for p/q and r/s wakes :

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

 $\theta _{-}(p/q)=\theta _{-}(1/4)=0.({\color {Blue}0001})$ 
 $\theta _{+}(p/q)=\theta _{+}(1/4)=0.({\color {Red}0010})$

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

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

Result is :

 $\theta _{-}(p/q,r/s)=\theta _{-}(1/4,1/5)=0.({\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Red}0010})$ 
 $\theta _{+}(p/q,r/s)=\theta _{+}(1/4,1/5)=0.({\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Red}0010}\ {\color {Blue}0001})$

One can check it using program Mandel by Wolf Jung :

The angle  69906/1048575  or  p00010001000100010010
has  preperiod = 0  and  period = 20.
The conjugate angle is  69921/1048575  or  p00010001000100100001 .
The kneading sequence is  AAABAAABAAABAAABAAA*  and
the internal address is  1-4-20 .
The corresponding parameter rays are landing
at the root of a satellite component of period 20.
It is bifurcating from period 4.
Do you want to draw the rays and to shift c
to the corresponding center?

3 angles

(1/2, 1/3, 1/4)

We go thru the list of angles from right to left

First compute (1/3,1/4) wake which will be used as a a new r/s wake :

  $\theta _{-}(1/3,1/4)=0.(001001001010)=\theta _{-}(r/s)$ 
  $\theta _{+}(1/3,1/4)=0.(001001010001)=\theta _{+}(r/s)$

After that compute 1/2 wake ( most left), which will be used as a p/q wake :

  $\theta _{-}(p/q)=\theta _{-}(1/2)=0.({\color {Blue}01})$ 
  $\theta _{+}(p/q)=\theta _{+}(1/2)=0.({\color {Red}10})$

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

digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$

Result is :

 $\theta _{-}(1/2,1/3,1/4)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$ 
 $\theta _{+}(1/2,1/3,1/4)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$

in plain text format ( for copy ) :

 theta_minus(1/2, 1/3, 1/4) = 0.(010110010110010110011001)
 theta_plus( 1/2, 1/3, 1/4) = 0.(010110010110010110010110)

One can check this wake in program Mandel by Wolf Jung using ray to point command ( Ctrl+e) :

The angle  5858713/16777215  or  p010110010110010110011001
has  preperiod = 0  and  period = 24.
The conjugate angle is  5858902/16777215  or  p010110010110011001010110 .
The kneading sequence is  ABABAAABABAAABABAAABABA*  and
the internal address is  1-2-6-24 .
The corresponding parameter rays are landing
at the root of a satellite component of period 24.
It is bifurcating from period 6.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3, 1/4, 1/5)

Input is a list :

(1/3, 1/4, 1/5)

We go thru the list of angles from right to left and divide list into 2 sublists :

 $p/q=1/3$ 

 $r/s=(1/4,1/5)$

First compute (1/4,1/5) wake which will be used as a a new r/s wake :

 $\theta _{-}(1/4,1/5)=0.(00010001000100010010)=\theta _{-}(r/s)$ 
 $\theta _{+}(1/4,1/5)=0.(00010001000100100001)=\theta _{+}(r/s)$

After that compute 1/3 wake ( most left), which will be used as a p/q wake :

  $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$ 
  $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

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

digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$

Result is :

 $\theta _{-}(1/3,1/4,1/5)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$ 
 $\theta _{+}(1/3,1/4,1/5)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$

One can check it using program Mandel by Wolf Jung :

The angle  164984615799661137/1152921504606846975  or  p001001001010001001001010001001001010001001001010001001010001
has  preperiod = 0  and  period = 60.
The conjugate angle is  164984615799689802/1152921504606846975  or  p001001001010001001001010001001001010001001010001001001001010 .
The kneading sequence is  AABAABAABAAAAABAABAABAAAAABAABAABAAAAABAABAABAAAAABAABAABAA*  and
the internal address is  1-3-12-60 .
The corresponding parameter rays are landing
at the root of a satellite component of period 60.
It is bifurcating from period 12.
Do you want to draw the rays and to shift c
to the corresponding center?

4 angles list

(1/2, 1/3, 1/4, 1/5)

Input is a list :

(1/2, 1/3, 1/4, 1/5)

so the internal addres should be :

1-2-6-24-120

One can not check it using program Mandel because it is limited to period 64.

We go thru the list of input angles from right to left and divide list into 2 sublists :

 $p/q=1/2$ 

 $r/s=(1/3,1/4,1/5)$

First compute (1/3, 1/4, 1/5) wake which will be used as a a new r/s wake :

 $\theta _{-}(1/3,1/4,1/5)=0.(001001001010001001001010001001001010001001001010001001010001)=\theta _{-}(r/s)$ 
 $\theta _{+}(1/3,1/4,1/5)=0.(001001001010001001001010001001001010001001010001001001001010)=\theta _{+}(r/s)$

After that compute 1/2 wake ( most left), which will be used as a p/q wake :

  $\theta _{-}(p/q)=\theta _{-}(1/2)=0.({\color {Blue}01})$ 
  $\theta _{+}(p/q)=\theta _{+}(1/2)=0.({\color {Red}10})$

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

digit 0 by repeating block (of length q, color blue) from $\theta _{-}(p/q)$
digit 1 by repeating block (of length q, color red) from $\theta _{+}(p/q)$

Result is ( to check !!!!) :

$\theta _{-}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$

$\theta _{+}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$

theta_minus(1/2, 1/3, 1/4, 1/5) = 0.(010110010110010110011001010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110)
theta_plus(1/2, 1/3, 1/4, 1/5)  = 0.(010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110010110010110010110011001)

One can check it visually using book program by Claude Heiland-Allen

size 640 360
view 53 -1.113644126576409e+00 2.5205986428803329e-01 3.9234950282896473e-04
ray_in 2000 .(010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110010110010110010110011001)
ray_in 2000 .(010110010110010110011001010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110)
text 53 -1.1152327443471231e+00 2.5276283972645397e-01 1/4
text 53 -1.1136201098499858e+00 2.5201617701965662e-01 1/5
text 53 -1.1152327443471231e+00 2.5276283972645397e-01 1/4
text 53 -1.1138472738947567e+00 2.5348331923684125e-01 24

References

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Fractals/Iterations in the complex plane/subwake

Algorithm = Douady tuning

Examples

2 angles

(1/3,1/2)

(1/3,1/3)

(1/3,1/4)

(1/3,3/4)

(1/4, 1/5)

3 angles

(1/2, 1/3, 1/4)

(1/3, 1/4, 1/5)

4 angles list

(1/2, 1/3, 1/4, 1/5)

See also

References