Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The Mandelbrot set is customarily defined as the set $M$ of all points $c\in\mathbb{C}$ such that the iterates of the function $z\mapsto z^2+c$, starting at $z=0$, remain bounded forever. Most very pretty depictions of the Mandelbrot set show $M$ as an intersection of an infinite sequence of sets $M_1\supset M_2\supset M_3\supset\cdots$, where the boundary of $M_i$ is the curve $|z_i(c)|=K$. Here $z_i(c)$ is the $i$th iterate of $z\mapsto z^2+c$, starting at $z=0$, and $K$ is some constant which guarantees that future iterates will escape. These curves $\partial (M_i)$ guide the viewer to see the increasingly intricate parts of the Mandelbrot set.
Each of these curves $\partial(M_i)$ is analytic and closed. They can thus be parametrized nicely with a trigonometric series. To be more specific, each boundary has a parametrization of the form
$$z(t)=\sum_{k=0}^\infty a_k\cos(kt)+i\sum_{k=0}^\infty b_k\sin(kt).$$
(In fact, since each boundary $\partial(M_i)$ is determined by a polynomial equation in the real and imaginary parts of $c$, I think each of these series should terminate. Correct me if I am wrong.) I would think that the limiting path should also have some nice parametrization with a trigonometric series. Is this limit the same for all $K$? If the limit is not the same for all $K$, then is there a limit as $K\rightarrow\infty$? What are the Fourier coefficients?
Best Answer
Lasse's answer expanded: Let $\psi$ be the map of the exterior of the unit disk onto the exterior of the Mandelbrot set, with Laurent series $$ \psi(w) = w + \sum_{n=0}^\infty b_n w^{-n} = w - \frac{1}{2} + \frac{1}{8} w^{-1} - \frac{1}{4} w^{-2} + \frac{15}{128} w^{-3} + 0 w^{-4} -\frac{47}{1024} w^{-5} + \dots $$ Then of course the boundary of the Mandelbrot set is the image of the unit circle under this map. However, this depends on the (not yet proved) local connectedness of that boundary. Here, for the coefficients $b_n$ there is no known closed form, but they can be computed recursively. Of course we put $w = e^{i\theta}$ and then this is a Fourier series.