Fractals/Iterations in the complex plane/pcheckerboard

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Name[edit]

  • Parabolic chessboard = parabolic checkerboard
  • zeros of qn

Algorithm[edit]

Star = Parabolic critical orbit with arms in different colors

Color depends on:

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



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

Steps[edit]

First choose:

  • choose target set, which is a circle with :
    • center in parabolic fixed point
    • radius such small that width of of exterior between components is smaller then pixel width
  • Target set consist of fragmnents 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)

Target set[edit]

  • 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 )

dictionary[edit]

  • 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

description[edit]

"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

"each yellow tile biholomorphically maps to the upper half plane, and 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]

Images[edit]

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]

code[edit]

0/1[edit]

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[edit]

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 also[edit]

references[edit]

  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