Name

Parabolic chessboard = parabolic checkerboard
zeros of qn - use only interior of Filled Julia set

Algorithm

Star = Parabolic critical orbit with 5 arms in different colors

Color depends on:

sign of imaginary part of numericaly aproximated Fatou coordinate (numerical explanation )
position of $z_{\infty }$ under $f^{p}$ 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

Steps

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 fragments of p components ( sectors). Divide each part of target set into 2 subsectors ( above and below critical orbit ) = binary decomposition

Steps:

take $z_{0}$ = 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

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 $\{z:z=f^{p}(z)\}$
- critical point $z_{cr}$
- one of 2 critical point preimages ( a or b ) $z=f^{-p}(z_{cr})$

How to compute preimages of critical point ?

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

dictionary

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

"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]

$\bigcup _{n=0}^{n\geq 0}f^{-n}(z_{cr})$

or

$\{z:f^{n}(z)=z_{cr}\}$

1/1

The parabolic chessboard for the polynomial  z + z^2  normalizing   $\psi _{att}(-1/2)=0$ 
* each yellow tile biholomorphically maps to the upper half plane
*  each blue tile biholomorphically maps to the lower half plane under  $\psi _{att}$ 
* The pre-critical points of  $z+z^{2}$  or equivalently the critical points of  $\psi _{att}$  are located where four tiles meet"^[5]

Images

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

on the real slice of Mandelbrot set
on the cusp of main cardioid
from main cardioid to period 2
between period 2 and 4 along 1/2 ray

on the boundary of main cardioid of Mandelbrot set
0/1 = 1/1
1/2
1/3

other polynomials
various number of star branches

orbits inside sepals
0/1 = 1/1
invariant curves inside main boxes
DLD

Examples :

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

code

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

0/1

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 basilica

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.