The dynamics of affine iteration is quite simple. When we iterate \(f(z)=az+b\) there just a few possibilities, assuming \(a\neq0\text{:}\)

1. \(|a| \lt 1\text{:}\) In this case, \(f\) has a unique, attractive fixed point, say \(z_0\text{.}\) If \(z_1\) is any complex number, then iteration of \(f\) from \(z_1\) will converge towards \(z_0\) at an exponential rate, i.e. like \(a^n\to0\text{.}\) If \(a\) is not a real number, then this convergence will spiral in to \(z_0\text{.}\)
2. \(|a| \gt 1\text{:}\) Again, \(f\) has a unique fixed point, say \(z_0\text{,}\) that is now repelling. If \(z_1\) is any complex number other than \(z_0\text{,}\) then iteration of \(f\) from \(z_1\) will diverge to \(\infty\) like \(a^n\to\infty\text{.}\) If \(a\) is not a real number, then this convergence will spiral out.
3. \(|a| = 1\text{:}\) Assuming that \(a\neq1\text{,}\) application of \(f\) represents pure rotation about some point. If \(a=1\) and \(b\neq 0\text{,}\) then \(f\) is a shift. In the final case that \(a=1\) and \(b=0\text{,}\) \(f\) is the identity function.

These observations characterize the dynamics of affine iteration completely. As a result, we will restrict further attention to more complicated maps.