Difference between revisions of "Fractals/Iterations in the complex plane/parabolic"

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File:Julia z2+0,25.png | from period 1 thru internal ray '''1/1'''; c = 0.25 = 1/4 the root of the main cardioid. Julia set is a '''cauliflower'''
 
File:Julia z2+0,25.png | from period 1 thru internal ray '''1/1'''; c = 0.25 = 1/4 the root of the main cardioid. Julia set is a '''cauliflower'''
File:Parabolic julia set c=-0.75.png | from period 1 to 2 thru internal ray '''1/2''' ; c=-0.75
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File:Parabolic julia set c=-0.75.png | from period 1 to 2 thru internal ray '''1/2''' ; c=-0.75 Julia set is a San Marco fractal<ref>[http://planetmath.org/encyclopedia/SanMarcoDragon.html planetmath : San Marco fractal]</ref>
 
File:Douady fat rabbit.jpg| from period 1 thru internal ray '''1/3''' ; Juia set is a '''fat Douady rabbit'''
 
File:Douady fat rabbit.jpg| from period 1 thru internal ray '''1/3''' ; Juia set is a '''fat Douady rabbit'''
 
File:Parabolic julia set c = -1.125 + 0.21650635094611*i.png | from period 2 thru 1/3
 
File:Parabolic julia set c = -1.125 + 0.21650635094611*i.png | from period 2 thru 1/3

Revision as of 17:09, 6 March 2012

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." ( Mark Braverman )[1]

Key words

  • parabolic chessboard
  • parabolic implosion
  • Ecalle cylinders or Ecalle-Voronin cylinders ( by Jean Ecalle [2])
  • Julia-Lavaurs sets
  • petal
  • The Leau-Fatou flower theorem
  • The horn map
  • Blaschke product
  • Inou and Shishikura's near parabolic renormalization

eggbeater dynamics

Hand Egg beater
Here is real model of what happens in parabolic case
It is a dynamic plane for fc(z)=z^2 + 1/4. It is zoom around parabolic fixed point z=0.5. Orbits of some points inside Julia set are shown (white points)

Physical model : the behaviour of cake when one uses eggbeater.

The mathematical model : a 2D vector field with 2 centers ( second-order degenerate points ) [3]


The field is spinning about the centers, but does not appear to be diverging.

Petal

There is no unified definition of petals.

Petal of a flower can be :

  • attracting
  • repelling


Attracting petal P is a :

  • domain (topological disc ) containing parabolic periodic point p in its boundary :

  • trap which captures any orbit tending to parabolic point [4]
  • set contained inside component of filled-in Julia set

0/1

How the target set is changing along an internal ray 0

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) [5]

C code :

 if (Zx>=0 && Zx <= 0.5 && (Zy > 0 ? Zy : -Zy)<= 0.5 - Zx) 
            { if (Zy>0) data[i]=200; // show petal
                     else data[i]=150;}

Gnuplot code :

reset
f(x,y)=  x>=0 && x<=0.5 &&  (y > 0 ? y : -y) <= 0.5 - x
unset colorbox
set isosample 300, 300
set xlabel 'x'
set ylabel 'y'
set sample 300
set pm3d map
splot [-2:2] [-2:2] f(x,y)

Flower

Flower with four petals
Critical orbit for internal angle 1/5 showing 5 directions

Sum of all petals creates a flower with center at parabolic periodic point.

Petals are symmetric with respect to the d-1 directions :

where :

  • d is a degree of complex quadratic map so d=2
  • l is from 0 to d-2

Petals can have different size.


Parabolic chessboard

Parabolic chessboard = parabolic checkerboard


See :

  • Tiles: Tessellation of the Interior of Filled Julia Sets by T Kawahira[6]
  • coloured califlower by A Cheritat [7]

Color points according to :[8]

  • the integer part of Fatou coordinate
  • the sign of imaginary part

Cauliflower

Cauliflower or broccoli :[9]

  • empty ( its interior is empty ) for c outside Mandelbrot set. Julia set is a totally disconnected (
  • filled cauliflower for c=1/4 on boundary of the Mandelbrot set. Julia set is a Jordan curve ( quasi circle).



Pleae note that :

  • size of image differs because of different z-planes.
  • different algorithms are used so colours are hard to compare

Bifurcation of the Cauliflower

How Julia set changes along real axis ( going from c=0 thru c=1/4 and futher ) :


Perturbation of a function by complex  :

When one add epsilon > 0 ( move along real axis toward + infinity ) there is a bifurcation of parabolic fixed point :

  • attracting fixed point ( epsilon<0 )
  • one parabolic fixed point ( epsilon = 0 )
  • one parabolic fixed point splits up into two conjugate repelling fixed points ( epsilon > 0 )

"If we slightly perturb with epsilon<0 then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling). "


See :

  • demo 2 page 9 in program Mandel by Wolf Jung

parabolic implosion

Fatou coordinate

Description at Hyperoperations Wiki

  • what we call "Abel function", they call it "Fatou coordinates".[10]
  • Fatou coordinates [11]
  • Shishikura perturbed Fatou coordinates [12]
  • Parabolic iteration [13][14]

Gallery

critical orbits for internal angle from 1/1 to 1/10

See also

References

  1. Mark Braverman : On efficient computation of parabolic Julia sets
  2. Jean Ecalle home page
  3. MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMSby P. Mardesic, R. Roussarie¤ and C. Rousseau
  4. IMPLOSION A MINI-COURSE by ARNAUD CHERITAT
  5. Program Mandel by Wolf Jung
  6. tiles by T Kawahira
  7. Coloured califlower by A Cheritat
  8. Applications of near-parabolic renormalization by Mitsuhiro Shishikura
  9. cauliflower at MuEncy by Robert Munafo
  10. new results from complex dynamics at Tetration Forum
  11. Fatou coordinate at Hyperoperations Wiki
  12. Shishikura perturbed Fatou coordinates
  13. Tetration Forum : Parabolic Iteration
  14. Tetration Forum : Parabolic Iteration, again
  15. planetmath : San Marco fractal
  16. Image : Nonstandard Parabolic by Cheritat
  17. Julia set of parabolic case in Maxima CAS