Fractals/Iterations in the complex plane/Fatou coordinate

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Fatou function[edit | edit source]

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  : "These are coordinates in which f looks like a translation." Małgorzata Stawiska[6]
  • maps petal to right half of plane in u coordinate.
  • unrolls invariant curvs ( orbits ) : maps "circles" to straight lines

Normalization[edit | edit source]

Fatou coordinate can be normalized = it maps critical point to zero  :[7]

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

Fatou coordinate[edit | edit source]

Fatou coordinate u :


Description at Hyperoperations Wiki

  • what we call "Abel function",[8] they call it "Fatou coordinates".[9]
  • Fatou coordinates [10][11]
  • Shishikura perturbed Fatou coordinates [12]

Computing[edit | edit source]

Programs[edit | edit source]

QFract[edit | edit source]

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 :


Search[edit | edit source]

References[edit | edit source]

  1. Tetration Forum : Parabolic Iteration
  2. Tetration Forum : Parabolic Iteration, again
  3. stackexchange : half-iterate-of-x^2 + c
  4. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda : Holomorphic Dynamic. January 13, 2000 | ISBN 0521662583 | ISBN 978-0521662581
  5. wiki : Abel%27s_equation
  6. Liouville theorem with parameters: asymptotics of certain rational integrals in differential fields by Małgorzata Stawiska
  7. Dynamics in one complex variable: introductory lectures by John W. Milnor, page 7-6
  8. wikipedia : Abel function
  9. new results from complex dynamics at Tetration Forum
  10. Minicourse "Analytic classification of germs of generic families unfolding a parabolic point
  11. Fatou coordinate at Hyperoperations Wiki
  12. Shishikura perturbed Fatou coordinates