Fractals/Iterations in the complex plane/pcheckerboard

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  • Parabolic chessboard = parabolic checkerboard
  • zeros of qn


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


  • 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

Target set[edit]

  • 2 triangles 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 )


  • 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


"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]


"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]


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]

See also[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