On the other hand, once we understand the behavior of an attractive fixed point, it's quite easy to extend that understanding to attractive orbits by examining the function $F=f^n\text{.}$ We'll also be able to use our analysis of attractive fixed points to understand repelling fixed points by considering $f^{-1}\text{.}$
The most subtle type of fixed point is a neutral fixed point, where $|f'(z_0)|=1\text{.}$ In this case, $f'(z_0)=e^{i\theta}$ and there are a variety of possibilities depending on the number theoretic properties of $\theta\text{.}$