Summary: Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if $X$ is a closed subscheme of $\{P}^n$ over a field, and $\phi \colon X \to X$ satisfies $\phi^* \mathscr{O}_X(1) \isom \mathscr{O}_X(d)$ for some $d \ge 2$, then there exists $r \ge 1$ such that $\phi^r$ extends to a morphism $\{P}^n \to \{P}^n$.

37P25, 37P55

self-map, closed point sieve