# Fractals/Iterations in the complex plane/Fatou coordinate

## Fatou function

Fatou function ${\displaystyle \Psi (z)}$ :[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 ${\displaystyle z\to z+1}$ : "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

${\displaystyle u=\Psi (z)}$

### Normalization

Fatou coordinate can be normalized = it maps critical point ${\displaystyle z=z_{cr}}$ to zero ${\displaystyle u=0}$ :[7]

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

Parabolic fixed point ${\displaystyle z_{f}}$ is mapped to point at infinity on Riemann sphere

${\displaystyle \Psi (z_{f})=\infty }$

## Fatou coordinate

Fatou coordinate u :

${\displaystyle u=\Psi (z)}$

where:

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]

# 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