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Fractals/Iterations in the complex plane/Fatou coordinate

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Fatou functionEdit

Fatou function   :[3]

  • is defined only inside petal ( attracting petal or repelling ), not on the whole neighbourhood of the fixed point
  • is a conformal function which satifies Abel's equation[4][5]
  • transforms f(z) to unit translation  
  • maps petal to right half of plane in u coordinate.
  • unrolls invariant curvs ( orbits ) : maps "circles" to straight lines



Fatou coordinate can be normalized = it maps critical point   to zero   :[6]


Parabolic fixed point   is mapped to point at infinity on Riemann sphere


Fatou coordinateEdit

Fatou coordinate u :


Description at Hyperoperations Wiki

  • what we call "Abel function"[7], they call it "Fatou coordinates".[8]
  • Fatou coordinates [9][10]
  • Shishikura perturbed Fatou coordinates [11]




QFract by INOU Hiroyuki and pictures

To build from the source code, you need :

Download source files from this page  :

First unpack the archive as follows

tar zcvf qfract-110725_2-src.tar.gz

Go to the program directory :

cd qfract-110725_2

and edit files :

  • Makefile,
  • config.h,
  • plugins/Makefile

to adjust your environment. For example in config.h change :

#define PLUGIN_PATH "/Users/inou/prog/qfract4/plugins"
#define COLORMAP_PATH "/Users/inou/prog/qfract4/colormaps"

for your own settings. Then to compile everything run from console :


To run the program from console :