Fractals/Iterations in the complex plane/tip misiurewicz
< Fractals
introduction[edit  edit source]

Shrub model of Mandelbrot set

parameter lane, wake 1/3 with rays landing on the principal Misiurewicz points
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]} "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)
 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."
 number
 binary
 decimal
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]

tip of the main antenna ( 1/2 wake)

1/3[edit  edit source]
Tips
 ftip = M_{2,1} = 0.01(0) = 1/4 = c = 0.228155493653962 +1.115142508039937 i
 ltip ??? (ToDo)
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[edit  edit source]
 c = 0.444556879255044 +0.409933108300984 i period = 0 // landing of 1/16
2/5[edit  edit source]
 c = 0.636754346582390 +0.685031297083677 i period = 0 // landing of 5/16
12/25[edit  edit source]
 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
mexrayout 100 0.7432918908524301 0.1312405523087976 8 1000 24 4
result :
.010101010101010101010100(1010)
which is
.01010101010101010101010(01)
See also[edit  edit source]
 What is the relation between period/preperiod of external angle of the ray landing on the tip and period/preperiod of the tip ( landing point) ?
All rays landing at the same periodic point have the same period: the common period of the rays is a (possibly proper) multiple of the period of their landing point; therefore one distinguishes: the ray period from the orbit period.^{[4]}
References[edit  edit source]
 20131002 islands in the hairs by Claude HeilandAllen
 20130201_navigating_by_spokes_in_the_mandelbrot_set by Claude HeilandAllen
 ↑ 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 MittagLeffler, The Royal Swedish Academy of Sciences, 7.