Fractals/Iterations in the complex plane/pcheckerboard

< Fractals < Iterations in the complex plane

Name

Parabolic chessboard = parabolic checkerboard

Algorithm

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

2 triangles 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]

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

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

"each yellow tile biholomorphically maps to the upper half plane, and each blue tile biholomorphically maps to the lower half plane under $\psi _{att}$ . 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

Examples :

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

references

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