Compressing the Mandelbrot set

Question

This question may not have a definitive answer. However, if someone is able to illuminate the topic for me, I would be very grateful.

The Mandelbrot set is the set obtained from the quadratic recurrence equation{1}:

$$ \begin{equation} z_{n+1}=z_n^2 + c \end{equation} $$

I'm sure most of you know what the graphical representation of the Mandelbrot set looks like, so I won't post a picture of it here.

Question

Have there been any attempts to derive the Mandelbrot set equation purely from it's graphical representation?

I would imagine that this would involve some sort of machine learning process which searches through program space trying to find a correct program with the smallest Kolmogorov complexity{2}.

What branch of mathematics works on solving this type of problem?

Thank you.

{1}: http://mathworld.wolfram.com/MandelbrotSet.html

{2}: http://en.wikipedia.org/wiki/Kolmogorov_complexity

Answer 1

In a certain sense the answer is yes -- look at Hubbard and Douady's work concerning "external angles" and "Hubbard trees". Modulo a conjecture about local path-connectedness I believe they have a very explicit topological model of the Mandelbrot set which in some sense is derived from a "picture" of it.

Answer 2

What graphical representation? The Mandelbrot set looks different at different resolutions. For a fixed resolution (and a fixed iteration threshold), you could try fractal image compression using iterated function systems, but I doubt the compression will be better than the definition. See this for an attempt.

One could say that the wonder of the Mandelbrot set is that so much information is compressed in such a simple definition. In that sense, I don't think you can compress the Mandelbrot set further.