This is a vague question, and I know nothing about this area.
We fix some $c\in\mathbb C$ and iterate the map $z\mapsto z^2+c$. This gives some filled Julia set, i.e. the set of points $z\in\mathbb C$ so that the orbit of $z$ is bounded. For example, for $c=0$ this is the closed unit disk. I'm wondering if anyone knows about any relationship between the area of the Julia set and the position of $c$ in the Mandelbrot set.