The dynamical chessboard of finite Blaschke-product

B3(z) = ((z+1/2)/(z/2+1))^3

It is inspired by second version of the paper "PARABOLIC AND NEAR-PARABOLIC RENORMALIZATION FOR LOCAL DEGREE THREE" by FEI YANG,

Here is original image extracted from the paper, see Figure 14 from page 52.: " the first column indicate the dynamical chessboards of f and B3"

Here is my image, it looks like 1/3 of original image:

Analysis

• B3 is a Blaschke product whose Julia set is the unit circle
• The point z = 1 is a 1-parabolic fixed point with two attracting petals
• the unit disk D and C∖D are two immediate basins of 1
• critical point: only one point z=-1/2
• poles: z = -2 (order 3 pole)
• zeros: z = -1/2 (order 3 zero)

Discussion

• math.stackexchange question: julia-set-of-finite-blaschke-product
• fractalforums.org: critical-points-of

Algorithm

Source code

Program:

• j.c - c console program
• j.sh - bash script
• Makefile - Makefile

Run: simply make from console

Output:

• png images
• j.txt - text output

other images

Git

create a new repository on the command line

echo "# The dynamical chessboard of finite Blaschke-product" >> README.md
git init
git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin git@github.com:adammaj1/dynamical-chessboard-of-Blaschke-product.git
git push -u origin main

Local repo

~/Dokumenty/julia_blaszke/b3/dynamical-chessboard-of-Blaschke-product/

Subdirectory

mkdir png
git add *.png
git mv  *.png ./png
git commit -m "move"
git push -u origin main

then link the images:

![](./png/n.png "description") 

gitm mv -f 

Remove a file/directory from a Git repository without deleting it from the local filesystem

git rm --cached myfile.log

Single directory and all it's content

git rm --cached -r ./png
git commit -m "png"
git push -u origin main

Rename directory

git add ./lcm/
git mv ./lcm ./png 
git commit -m "png"
git push -u origin main

Github

• GitHub Flavored Markdown Spec
• md cheat sheet
• CommonMark Spec
• Markdown parser