Fourier decomposition of the Mandelbrot set

Question

It is not clear that the boundary of the Mandelbrot set is an analytic curve, even though it is connected. Nevertheless, we can approximate the boundary with a curve by iterating a finite number of points at a given resolution $r$, and parameterize a curve from this

$$ f_r(t) = \frac{1}{L} (x(t) + iy(t)) $$

where $L$ is chosen such that $t$ ranges from 0 to 1. Let's just assume for the moment that $\lim_{r \rightarrow \infty} f_r(t) = M(t)$ is the boundary of the Mandelbrot set. What are the coefficients to the harmonic representation of $M$?

$$ \begin{align} M(t) &= \sum_{m=-\infty}^\infty a_m \exp(2\pi imt) \\ a_m &= \int_0^1 \exp(-2\pi mt) M(t) dt \end{align} $$

Answer 1

The boundary of the Mandelbrot set is most definitely not an analytic curve. However, the coefficients of the conformal map of the outside of the unit disk to the outside of the Mandelbrot set can be calculated recursively, based on results of Douady and Hubbard. Ewing and Schober calculated the first 240,000 coefficients and used them to estimate the area of the Mandelbrot set: http://link.springer.com/article/10.1007%2FBF01385497