introduction

Shrub model of Mandelbrot set
parameter lane, wake 1/3 with rays landing on the principal Misiurewicz points

key words definitions

Parts of the parameter plane

shrub
wake
limb
Misiurewicz point
- tip (point) = end point = branch tip = the tip of the spoke = terminal point of the branche^[1] = tip of the midget^[2] "A point in the Mandelbrot Set that is at the end of a filament (as opposed to a branch point); a point from which there is only one path to other points in the Mandelbrot Set."
  - The first tip = ftip = tip (end point) of the first and longest branch ( first when counting from the left to right)
  - the last tip = ltip = ( last when counting from the left to right)
number
- binary
- decimal

notation

preperiod

task

find Misiurewicz point
- preperiod and period
- c value
find angles of external ray that land on it

algorithms

Pastor

 "the external argument can be calculated as the limit of the arguments of the structural components of the branches 1, 11, 111,..., with periods 4, 5, 6,..., that is, the limit of .(0011), .(00111), .(001111),..., or the limit of .(0100), .(01000), .(010000), .... Hence, ftip(1/3) = .00(1) = .01(0), that are two equal values. " ^[3]

Claude

Method by Claude

Steps of the algorithm:

find angles of the wake
find angles of principal Misiurewicz point M
find angles of spoke's tips using: "The tip of each spoke is the longest matching prefix of neighbouring angles, with 1 appended"

1/3

3 angles landing on M:

  0.001(010) 
  0.001(100)
  0.010(001)

The tip of each spoke is the longest matching prefix of neighbouring angles, with 1 appended

  0.001(010) // 9/56 = 0.160(714285)
  0.0011    // ltip = 3/16 = 0.1875
  0.001(100) // 11/56 = 0.196(428571)
  0.01  // ftip = 1/4 = 0.25
  0.010(001) // 15/56 = 0.267(857142)

Check with program Mandel :

The angle  3/16  or  0011 has  preperiod = 4  and  period = 1. Entropy: e^h = 2^B = λ = 1.59898328
The corresponding parameter ray lands at a Misiurewicz point of preperiod 4 and period dividing 1. 
Do you want to draw the ray and to shift c to the landing point?

 c = -0.017187977338350  +1.037652343793215 i    period = 0

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
Entropy: e^h = 2^B = λ = 1.69562077
The corresponding parameter ray lands at a Misiurewicz point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?

 M_{2,1) = c = -0.228155493653962  +1.115142508039937 i

The angle  1/6  or  0p01 has  preperiod = 1  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 1 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?

 c = -0.000000000000000  +1.000000000000000 i    period = 10000

examples

1/2

tip of the main antenna ( 1/2 wake) $c=M_{1,1}$

   ${\begin{cases}0.(s_{-})=0.(01)={\frac {1}{3}}=0.(3)=wake\\0.s_{-}(s_{+})=0.01(10)={\frac {5}{12}}=0.41(6)=PrincipalMis=M_{2,2}\\0.01={\frac {1}{2}}=0.5=tip=M_{1,1}=c=-2\\0.s_{+}(s_{-})=0.10(01)={\frac {7}{12}}=0.58(3)=PrincipalMis=M_{2,2}\\0.(s_{+})=0.(10)={\frac {2}{3}}=0.(6)=wake\\\end{cases}}$

1/3

 ${\begin{cases}0.(001)_{2}={\frac {1}{7}}=0.(142857)_{10}=wake\\0.001(010)_{2}={\frac {9}{56}}=0.160(714285)=principleM\\0.001(100)_{2}={\frac {11}{56}}=0.196(428571)=principleM\\0.01_{2}={\frac {1}{4}}=0.25=ftip\\0.010(001)_{2}={\frac {15}{56}}=0.267(857142)=principleM\\0.(010)_{2}={\frac {2}{7}}=0.(285714)=wake\\\end{cases}}$

Tips

ftip = M_{2,1} = 0.01(0) = 1/4 = c = -0.228155493653962 +1.115142508039937 i
ltip ??? (ToDo)
- $0.0(01)={\frac {1}{6}}=0.1(6)_{10}=M_{2,1}=c=i$
- $0.0011(0)_{2}={\frac {3}{16}}=0.1875=M_{4,1}$

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
Entropy: e^h = 2^B = λ = 1.69562077
The corresponding parameter ray lands at a Misiurewicz point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?
c = -0.228155493653962  +1.115142508039937 i  

The angle  1/4  or  01 has  preperiod = 2  and  period = 1.
The corresponding dynamic ray lands at a preperiodic point of preperiod 2 and period dividing 1.
Do you want to draw the ray and to shift z to the landing point?
z = -0.228155493653962  +1.115142508039937 i

The angle  4/7  or  p100 has  preperiod = 0  and  period = 3. 
The dynamic ray lands at a repelling or parabolic point of period dividing 3.
Do you want to draw the ray and to shift z to the landing point?
z = -0.419643377607081  +0.606290729207199 i

The angle  1/8  or  001 has  preperiod = 3  and  period = 1.
The corresponding dynamic ray lands at a preperiodic point of preperiod 3 and period dividing 1.
Do you want to draw the ray and to shift z to the landing point?
z = 0.000000000159395  +0.000000000076028 i

The angle  3/16  or  0011 has  preperiod = 4  and  period = 1.
Entropy: e^h = 2^B = λ = 1.59898328 
The corresponding parameter ray lands at a Misiurewicz point of preperiod 4 and period dividing 1.
Do you want to draw the ray and to shift c to the landing point?
c = -0.017187977338350  +1.037652343793215 i    period = 0

The angle  1/6  or  0p01 has  preperiod = 1  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 1 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?
c = -0.000000000000000  +1.000000000000000 i    period = 10000

1/5

c = 0.444556879255044 +0.409933108300984 i period = 0 // landing of 1/16

2/5

c = -0.636754346582390 +0.685031297083677 i period = 0 // landing of 5/16

12/25

wake 12/25 of the main cardioid is bounded by the parameter rays with the angles
- 11184809/33554431 or p0101010101010101010101001 = 0.(0101010101010101010101001)
- 11184810/33554431 or p0101010101010101010101010

m-exray-out 100 -0.7432918908524301 0.1312405523087976  8 1000 24 4

result :

.010101010101010101010100(1010)

which is

.01010101010101010101010(01)

References

↑ Terminal Point by Robert P. Munafo, 2008 Mar 9.
↑ mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
↑ G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007. https://doi.org/10.1155/2007/45920
↑ H. Bruin and D. Schleicher, Symbolic dynamics of quadratic polynomials, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 7.

[1] Terminal Point by Robert P. Munafo, 2008 Mar 9.

[2] thoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?

[3] G. Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007. https://doi.org/10.1155/2007/45920

[4] H. Bruin and D. Schleicher, Symbolic dynamics of quadratic polynomials, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, 7.

[1]

[2]

[3]

[4]

Fractals/Iterations in the complex plane/tip misiurewicz

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