Consider the function f0(z)=z2 and an initial point z0. Let {zk} be the sequence of iterates of z0 generated by f0. That is, z1=f0(z0), z2=f0(z1), and so on.
(a) Show that, if ∣z0∣<1, the sequence {zk} converges to 0.
(b) Show that, if ∣z0∣>1, the sequence {zk} is unbounded.
(c) Show that, if ∣z0∣=1, the sequence {zk} either converges to 1 or oscillates around the unit circle. Give a simple criterion that you can apply to z0 that will reveal which of these two paths {zk} takes.