Each function \(f_c(z)=z^2+c\) has exactly one critical point at the origin. It turns out that the orbit of this critical point dominates the global dynamics of the iteration of \(f_c\text{.}\)

For the time being, we will content ourselves with a proof that the Julia set is totally disconnected when \(|c| \gt 2\) and sketch of the rest. The general idea is based on a process called inverse iteration. Given and function \(f:\mathbb C \to \mathbb C\) and a set \(S\in\mathbb C\) let

The set \(f^{-1}(S)\) is called the inverse image or pre-image of \(S\text{.}\) We define an initial approximation \(J_0\) and let \(J_n=f_c^{-n}(J_0)\equiv f_c^{-1}(J_{n-1})\text{.}\) As we will see, if the initial approximation is chosen correctly, the sequence of iterates will collapse down to filled Julia set.

The situation just described is more complicated when \(|c| \leq 2\text{.}\) If \(|c| \leq 2\) and the critical orbit escapes, then there is some first value of \(n\) with the property that \(|f_c^{n}(0)| \gt 2\text{.}\) We can then apply the inverse iteration idea to the disk of radius \(|f_c^{n}(0)|\) centered at the origin. The figure 8 configuration then appears after \(n\) iterates and the ideas of the proof are then applicable. This situation is illustrated in figure Figure 2(b).

If the critical orbit does not escape, then we can attempt inverse iteration with a disk \(D\) whose radius is larger than two. In this case, the figure 8 never forms so that \(f^{-n}(D)\) is a connected set for all \(n\text{.}\) As a result, the filled Julia set is a connected set since it's the intersection of a nested family of connected sets. This set is illustrated in Figure 2(c) and (d). In figure (c), \(f_c\) has an attractive orbit. In figure (d), \(f_c\) does not have an attractive orbit so that the sets \(f^{-n}(D)\) collapse down to have area zero. Since any attractive behavior must attract the critical orbit, one easy way to ensure the lack of attractive behavior is to chose \(c\) so that the orbit of zero lands exactly on a repelling orbit. In fact, one can prove that this always happens when zero is pre-periodic. In Figure 2(d), \(c=i\) and zero eventually maps to an orbit of period 2.