Fractals/Iterations in the complex plane/Fatou coordinate


 * Parabolic iteration

Fatou function
Fatou function $$ \Psi(z) $$ :
 * 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 $$z \to z+1$$ : "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

$$u = \Psi(z) $$

Normalization
Fatou coordinate can be normalized = it maps critical point $$z = z_{cr}$$ to zero $$ u = 0 $$ :

$$ \Psi(z_{cr}) = 0 $$

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

$$ \Psi(z_f) = \infty $$

Fatou coordinate
Fatou coordinate u :

$$u = \Psi(z) $$

where:
 * $$\Psi $$ is the Fatou function

Description at Hyperoperations Wiki
 * what we call "Abel function", they call it "Fatou coordinates".
 * Fatou coordinates
 * Shishikura perturbed Fatou coordinates

Computing

 * Fatou_coordinate for f(z)=z/(1+z)
 * Fatou_coordinate for f(z)=z+z^2
 * Fatou_coordinate for f(z)=z^2 + c

=Programs=

QFract
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 :

cd qfract-110725_2

and edit files :
 * Makefile,
 * config.h,
 * plugins/Makefile

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


 * 1) define PLUGIN_PATH "/Users/inou/prog/qfract4/plugins"
 * 2) define COLORMAP_PATH "/Users/inou/prog/qfract4/colormaps"

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

make

To run the program from console :

./qfract

=Search=
 * arxiv search: fatou+coordinate
 * math.stackexchange search: fatou+coordinate
 * mathoverflow search: fatou+coordinate

=References=