Section 3.1 An illustrative example
Let's begin by setting \(c=0\) so we can study what must be the simplest quadratic, namely \(f(z)=z^2\text{.}\) In fact, it's easy to obtain a closed form expression for the \(n^{\text{th}}\) iterate of \(f\text{,}\) namely: \(f^n(z)=z^{2^n}\text{.}\) In spite of this formula, iteration of \(f\) displays tremendously interesting dynamics indicative of much of what is to come. As we'll see, in fact, the dynamics of \(f\) naturally decomposes the complex plane into two regions: a stable region and an unstable region. A similar type of decomposition is possible for all polynomials and, even, for all rational functions. We'll come to call the stable region the Fatou set and the unstable region the Julia set.
We'll try to paint a static picture of the dynamics here but there is a dynamic and interactive illustration of the actual orbits of \(f\) here: https://www.marksmath.org/visualization/complex_square_iteration/.
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 3.1.1.
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.
The dynamics right on the unit circle \(C\text{,}\) by contrast are quite complicated. We'll show later that the dynamics on \(C\) displays all of the hallmarks of chaos. For the time being, we claim that \(f\) displays sensitive dependence to initial conditions. That is, given any point \(z_0\in C\) and any \(\varepsilon > 0\text{,}\) there is some \(w_0\in\mathbb C\) satisfying \(|z_0-w_0|<\varepsilon\) whose orbit is drastically different from the orbit of \(z_0\text{.}\) In particular, it is possible that the orbit of \(w_0\) tends to \(\infty\) while the orbit of \(z_0\) stays on the unit circle. The details are left as an exercise.