Fractals/Iterations in the complex plane/pcheckerboard

NameEdit

AlgorithmEdit

 
Star = Parabolic critical orbit with 5 arms in different colors

Color depends on:

  • sign of imaginary part of numerically approximated Fatou coordinate (numerical explanation )
  • position of   under   in relation to one arm of critical orbit star: above or below ( geometric explanation)


 
Attracting ( critical orbit) and repelling axes ( external rays landing on the parabolic fixed point ) divide niegbourhood of fixed point intio sectors

StepsEdit

First choose:

  • choose target set, which is a circle with :
    • center in parabolic fixed point
    • radius such small that width of exterior between components is smaller then pixel width
  • Target set consist of fragments of p components ( sectors). Divide each part of target set into 2 subsectors ( above and below critical orbit ) = binary decomposition

Steps:

  • take   = initial point of the orbit ( pixel)
  • make forward iterations
    • if points escapes = exterior
    • if point do not escapes then check if point is near fixed point ( in the target set)
      • if no then make some extra iterations
      • if is then check in what half of target set it ( binary decomposition)

How to find attracting petal ?Edit

Steps for parabolic basin

  • choose component with critical point inside
  • choose trap
  • decompose trap disk for binary decomposition = divide into 2 parts : above and beyond the critical orbit


Trap is a disk

  • inside component with critical point inside
  • trap has parabolic point on it's boundary
  • center of the trap is midpoint between last point of critical orbit and fixed point
  • radius of the trap is half of distance between fixed point and last point of critical orbit



Target set

  • circle ( for periods 1 and 2 in case of complex quadratic polynomial)
  • 2 triangles ( for periods >=3) described by :
    • parabolic periodic point for period p , find  
    • critical point  
    • one of 2 critical point preimages ( a or b )  

How to compute preimages of critical point ?Edit

  • (a,b)
  • (aa, ab)
  • (aaa,aab )
  • (aaaa, aaab )
  • (aaaaa, aaaab )

dictionaryEdit

  • The chessboard is the name of this decomposition of A into a graph and boxes
  • the chessboard graph
  • the chessboard boxes :" The connected components of its complement in A are called the chessboard boxes (in an actual chessboard they are called squares but here they have infinitely many corners and not just four). " [1]
  • the two principal or main chessboard boxes
  • trap = target set = attracting petal

Visualizing Structures with the Chessboard GraphEdit

  "An often used and very useful technique of visualization of ramified covers (and partial cover structures that are not too messy) consists in cutting the range in domains, often simply connected, along lines joining singular values, and taking the pre-image of these pieces, which gives a new set of pieces. The way they connect together and the way they map to the range give information about the structure." Arnaud Chéritat Near Parabolic Renormalization for Unicritical Holomorphic Maps by Arnaud Chéritat
 
Parabolic orbits insidse upper main chessboard box for f(z) = z^2 +0.25

descriptionEdit

"A nice way to visualize the extended Fatou coordinates is to make use of the parabolic graph and chessboard." [2]

Color points according to :[3]

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

Corners of the chessboard ( where four tiles meet ) are precritical points [4]

 

or

 

1/1Edit

The parabolic chessboard for the polynomial  z + z^2  normalizing   
* each yellow tile biholomorphically maps to the upper half plane
*  each blue tile biholomorphically maps to the lower half plane under  
* The pre-critical points of   or equivalently the critical points of   are located where four tiles meet"[5]

ImagesEdit

Click on the images to see the code and descriptions on the Commons !






Examples :

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

codeEdit

For the internal angle 0/1 and 1/2 critical orbit is on the real line ( Im(z) = 0). It is easy to compute parabolic chesboard because one have to check only imaginary part of z. For other cases it is not so easy

0/1Edit

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 ) [8][9]

To see effect :

  • run Mandel
  • (on parameter plane ) find parabolic point for angle 0, which is c=0.25. To do it use key c, in window input 0 and return.

C code :

  // in function uint mndlbrot::esctime(double x, double y)
  if (b == 0.0 && !drawmode && sign < 0
      && (a == 0.25 || a == -0.75)) return parabolic(x, y);
 // uint mndlbrot::parabolic(double x, double y)
 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)

1/2 or fat basilicaEdit

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) [10] To see effect :

  • run Mandel
  • (on parameter plane ) find parabolic point for angle 1/2, which is c=-0.75. To do it use key c, in window input 0 and return.

C code :

  // in function uint mndlbrot::esctime(double x, double y)
  if (b == 0.0 && !drawmode && sign < 0
      && (a == 0.25 || a == -0.75)) return parabolic(x, y);
 // uint mndlbrot::parabolic(double x, double y)
  if (A < 0 && x >= -0.5 && x <= 0 && (y > 0 ? y : -y) <= 0.3 + 0.6*x)
      {  if (j & 1) return (y > 0 ? 65282u : 65290u);
         else return (y > 0 ? 65281u : 65289u);
      }


See alsoEdit

referencesEdit

  1. Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
  2. About Inou and Shishikura’s near parabolic renormalization by Arnaud Cheritat
  3. Applications of near-parabolic renormalization by Mitsuhiro Shishikura
  4. Complex Dynamical Systems by Robert L. Devaney, page
  5. Antiholomorphic Dynamics: Topology of Parameter Spaces and Discontinuity of Straightening by Sabyasachi Mukherjee
  6. tiles by T Kawahira
  7. checkerboards by A Cheritat
  8. commons:Category:Fractals created with Mandel
  9. Program Mandel by Wolf Jung
  10. Program Mandel by Wolf Jung
  11. Oleg Ivrii (Tel Aviv University), "Shapes of trees"