Fractals/Iterations in the complex plane/tip misiurewicz

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introduction[edit | edit source]


key words definitions[edit | edit source]

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]
      • The first tip = ftip
      • the last tip = ltip

notation[edit | edit source]

task[edit | edit source]

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

algorithms[edit | edit source]

Pastor[edit | edit source]

 "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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

1/2[edit | edit source]

References[edit | edit source]

  1. Terminal Point by  Robert P. Munafo, 2008 Mar 9.
  2. mathoverflow 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