# Fractals/Iterations in the complex plane/Fatou coordinate

## 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 

# Programs

## QFract

To build from the source code, you need :

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

#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