As it turns out, the Julia set of \(f(z) = z^2 + c\) is either "in on piece" or "totally dusty". This leads to the definition of a Mandelbrot set \(M\), which is the set of all parameters \(c \in \mathbb{C}\) for which such Julia set is connected:

\[ \begin{aligned} M = \{ c \in \mathbb{C}: J(z^2 + c) \text{ is connected} \} \end{aligned} \]

Here are two amazing theorems:

\( J(z^2 + c)\) is connected if and only if 0 does not belong to \(A(\infty)\) !

In other words, \(\{f^n(0)\}\) remains bounded under iteration. Furthermore,

A complex number \(c\) belongs to \(M\) if and only if \(\lvert f^n(0) \rvert \le 2 \) for all \(n \ge 1\) !

where \(f(z) = z^2 + c\).

You can find more info at Analysis of a Complex Kind.