# Mandelbrot set's border in parametric form

I've post this question just because I'm curious,

Mandelbrot set is defined as: $z_{n+1} = z^2_n + c$, if $n \rightarrow \infty$ and it doesn't diverge we get the border. This border is unlimited and it's direction isn't definite, so I think it's a problematic.

My question is if we can write the border in a parametric way like this: $\left \lbrace \begin{array}{l} x = x(t) \\ y = y(t) \end{array} \right.$ , if we need some extra variabile to approximate the system or it's impossible to write it in this form. ($z \in \mathbb{C}$ so we can decompose it in: $z = x + yi$)

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Good question! I'm quite sure that there is no easily computable parameterization of the boundary of the Mandelbrot set. But whether there is any (not necessarily nice) continuous parameterization of the boundary is less clear. It seems likely that the proof that the Mandelbrot set is connected can be adapted to show that there is, but it doesn't follow from the naked fact that the set is connected that it also has a continuously parameterizable contour. –  Henning Makholm Oct 4 '12 at 21:05
Yes, we can. Probably. More later... –  GEdgar Oct 4 '12 at 22:09
Here is the answer from MathOverflow last December ... mathoverflow.net/questions/48642/… –  GEdgar Oct 5 '12 at 0:34
@GEdgar: Thank you very much, the note has been very useful. Even if I don't understand how can I use Fourier series, I've seen how to get coefficients (the most important thing), I report the link taken from a comment. –  Blex Oct 5 '12 at 13:27