Fractals/Iterations in the complex plane/Fatou coordinate
Fatou function :
- 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
- transforms f(z) to unit translation : "These are coordinates in which f looks like a translation." Małgorzata Stawiska
- 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 :
Parabolic fixed point is mapped to point at infinity on Riemann sphere
Fatou coordinate u :
Description at Hyperoperations Wiki
- what we call "Abel function", they call it "Fatou coordinates".
- Fatou coordinates 
- Shishikura perturbed Fatou coordinates 
QFract by INOU Hiroyuki and pictures
To build from the source code, you need :
- Qt 4.5 (or later?) from http://qt.nokia.com/ and install.
- Boost C++ Libraries (http://www.boost.org/).
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 :
and edit files :
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 :
- ↑ Tetration Forum : Parabolic Iteration
- ↑ Tetration Forum : Parabolic Iteration, again
- ↑ stackexchange : half-iterate-of-x^2 + c
- ↑ S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda : Holomorphic Dynamic. January 13, 2000 | ISBN 0521662583 | ISBN 978-0521662581
- ↑ wiki : Abel%27s_equation
- ↑ Liouville theorem with parameters: asymptotics of certain rational integrals in differential fields by Małgorzata Stawiska
- ↑ Dynamics in one complex variable: introductory lectures by John W. Milnor, page 7-6
- ↑ wikipedia : Abel function
- ↑ new results from complex dynamics at Tetration Forum
- ↑ Minicourse "Analytic classification of germs of generic families unfolding a parabolic point
- ↑ Fatou coordinate at Hyperoperations Wiki
- ↑ Shishikura perturbed Fatou coordinates