# Fractals/Iterations in the complex plane/pcheckerboard

< Fractals

## Contents

# Name[edit]

- Parabolic chessboard = parabolic checkerboard
- zeros of qn

# Algorithm[edit]

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)

## Steps[edit]

- 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 )

# 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]}

# See also[edit]

- Checkerboard in Hyperbolic tilings by User:Tamfang : images and Python code
- https://plus.google.com/110803890168343196795/posts/Eun6pZVkkmA
- shadertoy: Orbit trapped julia Created by maeln in 2016-Jan-19
- Holomorphic checkerboard by etale_cohomology

# references[edit]

- ↑ Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
- ↑ About Inou and Shishikura’s near parabolic renormalization by Arnaud Cheritat
- ↑ Applications of near-parabolic renormalization by Mitsuhiro Shishikura
- ↑ Complex Dynamical Systems by Robert L. Devaney, page
- ↑ Antiholomorphic Dynamics: Topology of Parameter Spaces and Discontinuity of Straightening by Sabyasachi Mukherjee
- ↑ tiles by T Kawahira
- ↑ checkerboards by A Cheritat