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English: Numerical aproximation of Julia set. Map is complex quadratic polynomial. Parameter c is on boundary of main cardioid of period one component of Mandelbrot set with rotational number ( internal angle ) = Golden Mean. Made with modified inverse iteration method MIIM/J with hit limit. It contains Siegel disc
Polski: Numeryczna przybliżenie zbioru Julii
Date
Source Own work
Author Adam majewski
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Summary

This image shows main component of Julia set and its preimages under complex quadratic polynomial[1] up to level 100008 ( new code ) or 25 ( old code).

Main component of Julia set :

  • contains Siegel disc ( around fixed point )
  • its boundary = critical orbit
  • it is a limit of iterations for every other component of filled Julia set

computing c parameter

Parameter c is on boundary of period one component of Mandelbrot set ( = main cardioid ) with combinatorial rotation number ( internal angle ) = inverse Golden Mean[2] .


As a result one gets function describing relation between parameter c and internal angle  :


One can compute boundary point c

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf Jung[3]

phi *= (2*PI); // from turns to radians
cx = 0.5*cos(phi) - 0.25*cos(2*phi); 
cy = 0.5*sin(phi) - 0.25*sin(2*phi); 

Algorithm

  • draw critical orbit = forward orbit of critical point ( "Iterates of critical point delineate a Siegel disc"[4])
  • for each point of critical orbit draw all its preimages up to LevelMax if Hit<HitLimit

Critical orbit in this case is : dense [5]( correct me if I'm wrong ) in boundary of component of filled-in Julia set containing Siegel disc.[6]

Mathemathical description

Description [7]

Quadratic polynomial [8] whose variable is a complex number

contains invariant Siegel disc  :

Boundary of Siegel disc

  • contains critical point  :
  • is a Jordan curve
  • is invariant under quadratic polynomial  :
  • is a closure of forward orbits of critical points

Julia is build from preimages of boundary of Siegel disc ( union of copies of B meeting only at critical point and it's preimages ):[9]

Here maximal level is not infinity but finite number :

jMax = LevelMax = 100008;

Code efficiency

Recursion

This code uses recursion[10] inside DrawPointAndItsInverseOrbit function so its runing time is (if I'm not wrong ) exponential [11] For level about 25 old code needs 24 hours and for level LevelMax new code needs 37m50.959.

Number of points

Level New components All components New points All points
0 1 1 NrOfCrOrbitPoints NrOfCrOrbitPoints
1 1 2 NrOfCrOrbitPoints 2*NrOfCrOrbitPoints
2 2 4
3 4 8
4 8 16
... ... ... ...
100008
j

Components gets smaller as level increases ( but with some varioations) so nr of points to draw can be diminished.

C src code

Src code was formatted with Emacs

New c code : MIIM/J ( hit limit )

Postprocessing

Using Image Magic :

  • edge detection
  • resize
  • convert to black and white image
  • inversion
convert L100008.pgm -convolve "-1,-1,-1,-1,8,-1,-1,-1,-1"  -resize 1500x1100 -threshold 5% -negate 1500e6n95.png

Compare with

References

  1. complex quadratic polynomial
  2. wikipedia : Golden ratio
  3. Mandel: software for real and complex dynamics by Wolf Jung
  4. Complex Dynamics by Lennart Carleson and Theodore W. Gamelin. Page 84
  5. Dense set in wikipedia
  6. Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201
  7. A. Blokh, X. Buff, A. Cheritat, L. Oversteegen The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory and Dynamical Systems , 30 (2010), #1, 51-65,
  8. wikipedia : Complex quadratic polynomial
  9. Building blocks for Quadratic Julia sets by : Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen Journal: Trans. Amer. Math. Soc. 351 (1999), 1171-1201
  10. wikipedia : Recursion
  11. Big O notation in wikipedia

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13 November 2011

File history

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Date/TimeThumbnailDimensionsUserComment
current16:39, 17 November 2011Thumbnail for version as of 16:39, 17 November 20111,500 × 1,100 (27 KB)Soul windsurferNew version : new code and new conversion
08:38, 16 November 2011Thumbnail for version as of 08:38, 16 November 20111,500 × 1,100 (39 KB)Soul windsurfer25 level - 2 days !!! ( I must improve program)
13:28, 13 November 2011Thumbnail for version as of 13:28, 13 November 20111,500 × 1,100 (39 KB)Soul windsurfer

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