Summary

Licensing

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https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Summary

Components

From math point of view there are :

• Julia set^[1] ( common boundary of Fatou components ) ${\displaystyle J(f)\,=\partial K(f)=\partial A_{f}(\infty )}$
  
• Fatou set^[2] ${\displaystyle F(f)=\mathbb {C} \setminus J(f)}$

Fatou set consist of components ( subsets) :

• attracting components ( here exterior of Julia set )
• parabolic compoent ) here interior of Julia set )

From programming point of view ( because of numerical methods and limited precision ) there are 3 components of image :

• known exterior = escaping set, all pixels that escape to infinity ${\displaystyle A_{f}(\infty )=\{z:abs(z)>2\}}$
• known interior = attracting set, all pixels which are attracted by alfa fixed point after limited iterations ( Max iter )
• unknown pixels, all pixels which do not escape or not attracted by alfa fixed point after limited iterations ( Max iter )

Data.text file

# maxiter ExteriorPixels InteriorPixels UnknownPixesl
 0 0.000000 0.000234  0.999766
 1 0.328092 0.000477  0.671431
 2 0.547592 0.000575  0.451833
 3 0.640913 0.000653  0.358433
 4 0.687845 0.000693  0.311461
 5 0.714504 0.000721  0.284775
 6 0.730709 0.000741  0.268549
 7 0.741129 0.000757  0.258113
 8 0.748003 0.000765  0.251232
 9 0.752817 0.000772  0.246411
 10 0.756136 0.000776  0.243088
 11 0.758540 0.000780  0.240680
 12 0.760269 0.000785  0.238945
 13 0.761567 0.000788  0.237645
 14 0.762573 0.000788  0.236639
 15 0.763303 0.000789  0.235908
 16 0.763921 0.000789  0.235289
 17 0.764345 0.000791  0.234864
 18 0.764708 0.000792  0.234500
 19 0.765025 0.000792  0.234183
...

Gnuplot src code

set terminal postscript portrait enhanced mono dashed lw 1 "Helvetica" 14     
set output "data1.ps"
set title "Points on dynamical plane "
set ylabel "ratio"
set xlabel "iteratio max"
#set xrange [0:150]
# define styles 
#set style data lines
set style line 1 lc rgb "green" lt 1 lw 5 pt 1 ps 1.5   # 
set style line 2 lc rgb "blue"  lt 1 lw 5 pt 1 ps 1.5   # 
set style line 3 lc rgb "red"   lt 1 lw 5 pt 1 ps 1.5   # 
# plot "data.txt" using 1:2 title 'exterior' lc 1,  "data.txt" using 1:3 title 'interior' lc 2, "data.txt" using 1:4 title 'unknown' lc 3
# plot "data.txt" using 1:2 title 'exterior' with linespoints ls 1, "data.txt" using 1:3 title 'interior' with linespoints ls 2, "data.txt" using 1:4 title 'unknown' with linespoints ls 3
# To reuse the last filename we can just type ""
# plot "data.txt" using 1:2 title 'exterior' with linespoints ls 1, "" using 1:3 title 'interior' with linespoints ls 2, "" using 1:4 title 'unknown' with linespoints ls 3
plot "data.txt" using 1:2 title 'exterior' with lines ls 1, "" using 1:3 title 'interior' with lines ls 2, "" using 1:4 title 'unknown' with lines ls 3

References

1. ↑ Julia set in wikipedia
2. ↑ Classification of Fatou components in wikipedia