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 :

#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 :

make

To run the program from console :

./qfract

Search

• arxiv search: fatou+coordinate
• math.stackexchange search: fatou+coordinate
• mathoverflow search: fatou+coordinate