Artin–Mazur zeta function

Summary

The Artin–Mazur zeta function of a map packs the numbers of periodic points of all periods into one power series. For a map $f$ with finitely many fixed points of each iterate it is $\exp \sum_{n \ge 1} N_n z^n / n$, where $N_n$ is the number of fixed points of $f^n$. For many hyperbolic systems it is a rational function, and it then records all periodic orbit counts by finitely many numbers. For the Frobenius map of a variety over a finite field it is the local zeta function of the variety.