# Deriving convergence region of iterative formula

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.

• I believe that curve is one component of $|f(z)| = |z|$. Initial points outside that curve map to larger numbers, which is why you get the divergence. – Mark McClure Mar 11 '16 at 13:09
• Thank you, that seems the be the curve I am looking for, see my edit... – Douwe66 Mar 11 '16 at 15:51
• Yes, that agrees with my image. I guess that $|F(z)|>|z|$ for points inside the interior curve but they still don't exit the region bounded by the exterior curve. – Mark McClure Mar 11 '16 at 15:56