Compressing the Mandelbrot set

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

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As far as I am aware, there is no computationally precise way of encoding the Mandlebrot set except for the definition of the Mandlebrot set (although there are slightly different theoretical descriptions which are easily equivalent).  You can't compress a collection of data if you don't at least have some finite yet inefficient way of representing it.  The "graphical representation" of the Mandlebrot set is not actually such a representation, it is just a series of approximations. – Aaron May 30 '11 at 16:39
– lhf May 30 '11 at 16:44

2 Answers

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.

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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.

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Regarding your second paragraph, that is indeed my point. The Mandelbrot set equation is the simplest compression of the Mandelbrot set. My question is, are there techniques for deriving the Mandelbrot set equation from the Mandelbrot set of points. – Ncarlson May 30 '11 at 16:51
@NCarlson You can't derive anything from the Mandelbrot set of points unless you have some way of deciding what it means to be in the Mandlebrot set of points.  If we throw the definition away, what would be the starting point?  For a slightly different example, suppose that you wanted to characterize the prime numbers, but you wanted to ignore that they were the prime numbers.  You could ask "What is something that generates this set?", but if you throw away the definition of prime, how do you describe the set in the first place?  It is infinite. – Aaron May 30 '11 at 16:59
@Aaron, the goal is to find the Mandelbrot set equation from the Mandelbrot set of points. In the case of primes, given a finite the finite set {2, 3, 5, 7, 11, 13}, there are numerous programs which will produce this set. A polynomial approximation would be one program. However, if our finite set of primes has 1,000,000 elements, then a polynomial approximation would be much more complex than a program which stated "A set of natural numbers where each element is only divisible by itself and 1". – Ncarlson May 30 '11 at 17:20
@Aaron, " if you throw away the definition of prime, how do you describe the set in the first place?". We assume that a (finite) set is given to us as some sort of binary input. – Ncarlson May 30 '11 at 17:22
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