Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$.
What are some examples of surface automorphisms $g$ with no periodic points?
For example, $S$ can be an abelian variety and $g$ could be translation by some general point $\tau\in S$. Are there others, say on a rational surface, or a K3 surface?