Let's suppose that \(|z_0|\lt1\text{,}\) i.e. \(z_0\) is in the interior of the unit circle in the complex plane. Then, since \(f^n(z_0)=z_0^{2^n}\text{,}\) it's easy to see that the iterates of \(z_0\) converge to zero. In addition, the argument of the iterate doubles with each application of \(f\text{.}\) As a result, there is a spiral effect as well.

If \(|z_0|\gt1\text{,}\) i.e. \(z_0\) is in the exterior of the unit circle in the complex plane, then the iterates spiral out away from the circle and towards infinity. The dynamics of \(f\) is shown for several choices of \(z_0\) in figure Figure 1.

Let \(F\) denote the complement of the unit circle. Thus, \(F\) is an open set consisting of two, disjoint parts - the interior of the unit circle and the exterior. The dynamics of \(f\) on \(F\) are stable in a concrete sense. Suppose that \(z_0\in F\text{.}\) Then, there is a disk \(D\) centered at \(z_0\) whose radius is small enough that \(D\subset F\text{.}\) This means that \(D\) lies entirely within the interior of the unit circle or within the exterior. Either way, the long term behavior of every point in \(D\) is the same as the long term behavior of \(z_0\text{.}\) This is exactly what we mean by stability in \(F\text{:}\) changing the initial input a little bit does not change the long term behavior. This is exactly the opposite of sensitive dependence on initial conditions.