Identifying unordered set of complex data as originating from a particular Mandelbrot process?

Question

Here is an old question which probably originated back in my high school days.

When iterating the Mandelbrot

$$z_{i+1} = {z_i}^2+c$$

we get a sequence of complex numbers. If we plot these in the complex plane, we can get lots of different looks:

Now if we do not have ordered sequence, like a "time series" of these $z_k$ : $\{z_k,z_{k+1},\cdots ,z_n\}$ but instead just a plot like this with a unordered set of positions, can we calculate "backwards" what the original $c$ is?

Answer 1

You are drawing critical orbit = forward orbit of a critical point.

You ask : can we read/approximate the parameter c from the graph/image of critical orbit ?

I think that it is possible to some extent:

• first image: c is probably from interior of Mandelbrot set main cardioid (= internal radius < 1), with internal angle smaller then n/8 ( 8 arms , turning clockwise), for example c = 0.345805039440922 +0.097755675960631 i

• last image looks like a periodic orbit with period = 14 , so it maybe center of hyperbolic component with period 14, like for example c = 0.295453572980359 +0.020451548527456 i

• middle image looks like Siegel disc

But remember that because of slow dynamics and numerical errors first n-points may look different then images with more points.

The next questions could be :

• how critical orbit is changing when c is moving on the parameter plane?
• how to move on the parameter plane ?

HTH