# Name

• Parabolic chessboard = parabolic checkerboard
• zeros of qn

# Algorithm

Star = Parabolic critical orbit with arms in different colors

Color depends on:

• sign of imaginary part of Fatou coordinate ( )
• position of ${\displaystyle z_{\infty }}$ under ${\displaystyle f^{p}}$ 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

• 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

• 2 triangles described by :
• parabolic periodic point for period p , find ${\displaystyle \{z:z=f^{p}(z)\}}$
• critical point ${\displaystyle z_{cr}}$
• one of 2 critical point preimages ( a or b ) ${\displaystyle z=f^{-p}(z_{cr})}$

# 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]

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

or

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

"each yellow tile biholomorphically maps to the upper half plane, and each blue tile biholomorphically maps to the lower half plane under ${\displaystyle \psi _{att}}$. The pre-critical points of ${\displaystyle z+z^{2}}$ or equivalently the critical points of ${\displaystyle \psi _{att}}$ are located where four tiles meet"[5]

# Images

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]