I am attempting an exercise in Milnor's Dynamics in One Complex Variable which I have slightly rephrased below:
Let $f \in \mathbb{C}(z)$ be a rational function of degree $d \geq 2$. Prove: $0, \infty$ are exceptional points for $f$ if and only if $f(z) = \alpha z^n$, where $\alpha \neq 0, n = \pm d$. Deduce that the exceptional set $\mathcal{E}(f)$ (defined to be the set of points whose grand orbit under $f$ is finite), is nonempty if and only if $f$ is conjugate via a Möbius transformation to a polynomial or $z \mapsto 1/z^d$.
Clearly, $0, \infty$ are exceptional points for $f(z) = \alpha z^n$ for any $\alpha > 0$ and $n$ either $d$ or $-d$, but I can't prove that these are the only cases for which those two points are exceptional, nor can I show (assuming this) the conclusion, the criterion for when the exceptional set of a rational function is finite. Thank you for your help.