Fractals/Iterations in the complex plane/Fatou coordinate

< Fractals < Iterations in the complex plane

Description

Fatou function $\Psi (z)$ :^[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 $z\to z+1$
maps petal to right half of plane in u coordinate.
unrolls invariant curvs ( orbits ) : maps "circles" to straight lines

$u=\Psi (z)$

Fatou coordinate can be normalized = it maps critical point $z=z_{cr}$ to zero $u=0$ :^[6]

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

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

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

Fatou coordinate u :

$u=\Psi (z)$

Description at Hyperoperations Wiki

To build from the source code, you need :

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 :

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

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