Mandelbrot set

Description

For $c \in \C$ let $f_c(z) = z^2 + c$, the quadratic family. The Mandelbrot set is $$M = \set{c \in \C : \text{the sequence } f_c^n(0),\ n \ge 0, \text{ is bounded}}.$$ Here $0$ is the only critical point of $f_c$. Check of a few points:

1. $c = 0$: the orbit of $0$ is $0, 0, 0, \ldots$, so $0 \in M$.
2. $c = -1$: the orbit is $0, -1, 0, -1, \ldots$, so $-1 \in M$.
3. $c = i$: the orbit is $0, i, -1 + i, -i, -1 + i, -i, \ldots$, so $i \in M$.
4. $c = 1$: the orbit is $0, 1, 2, 5, 26, \ldots$, which is unbounded, so $1 \notin M$.

If some $\abs{f_c^n(0)} > 2$, the orbit tends to $\infty$. So $M$ is the set of $c$ with $\abs{f_c^n(0)} \le 2$ for all $n$, and $M$ is a compact subset of the disc $\abs{c} \le 2$.