On a previous qualifying exam, one of the questions was
Let $\Sigma_3$ be the closed orientable surface of genus 3. Suppose $\mathbb{Z}_3$ acts on $\Sigma_3$. Show that there are at least 2 fixed points.
The only theorem I know related to fixed points, the Lefschetz fixed point theorem, only shows existence of fixed points. Maybe covering space theory could be of use. If $\mathbb{Z}_3$ acts by rotation, it is clear that there are two fixed points, but I don't know about other possible actions.
Think about the complement of the fixed set. It carries a free $\Bbb Z/3$ action; thus (because we can quotient by this action and Euler char is multiplicative under finite covers) $\chi(\Sigma_3 \setminus F)$ is divisible by 3. Furthermore, if $F$ is finite, $\chi(\Sigma_3 \setminus F) = -4-|F|$. This is only possible if $|F|$ is 2 mod 3, and in particular must be at least 2!