A year ago I asked this question about fractal icons, however I didn't get any wiser yet. Now I am trying to understand the convergence of a simplified version of the fractal, to learn more about the mathematics behind it.
As an example I took the iterative formula shown below:
$$F(z) = \{2.5 -2.5 z \bar{z} \}z + 0.5 \bar{z}^{2} $$
After a lot of iterations the fractal below is plotted using a color map.
Starting points inside the green line in the image above result in iterations, with which I have drawn the fractal. Starting points outside the green line result in $F(z)$ exploding and not creating a fractal.
Is it possible to derive the formula for this green line using the fractal formula?
As a starting point I looked at the roots of the function $F(z)$, they all seem to lay on the green line, but I'm not sure where to go from there...
Edit: Suggestion by @Mark McClure
Using matlab I have plotted the points for which $|F(z)| < |z|$, this shows that there are 2 lines for which $|F(z)| = |z|$. The outer line seems to be the one I am looking for.
