 Solution

Suppose first that $|E|\ge 3$. Complete invariance implies that every iterate maps $\hat{\mathbb{C}}\setminus E$ into itself. Hence the family $\{f^n\}$ omits the same three points of $E$ on this open set. By Montel theorem,so

If $E$ consists of one or two points, complete invariance makes $f$ permute those points and makes every preimage of them remain in $E$. Some iterate fixes each point and is totally ramified there. In a local coordinate it therefore has the form $w\mapsto a{w}^m+O(w^{m+1})$ with $m\ge 2$, so the point is superattracting for that iterate and belongs to the Fatou set. This proves the completely invariant closed set of a rational map dichotomy.