Section3.2\(z^2\text{:}\) The unit circle
The simplest quadratic, obviously, is \(f(z)=z^2\text{.}\) 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://goo.gl/xBvFeo.