---
tag: 2A5Z21CE4P
kind: concept
name: Artin–Mazur zeta function
msc: ["37C30", "37C25"]
sources:
  - url: https://en.wikipedia.org/wiki/Artin%E2%80%93Mazur_zeta_function
same_as:
  wikidata: Q713051
  wikipedia: Artin–Mazur zeta function
---

## Summary

The Artin–Mazur zeta function of a map packs the numbers of [[periodic points|T75CZ5H47N]] 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.

## Explanations

### L1 human

Counting periodic points of a map gives a list of numbers $N_1, N_2, N_3, \dots$, one for each period.
The zeta function turns the whole list into a single function of one variable.
For the doubling map $z \mapsto z^2$ on the unit circle, $N_n = 2^n - 1$, and the zeta function turns out to be the simple fraction $(1 - z)/(1 - 2z)$.
The pole at $z = 1/2$ shows that the number of periodic points grows like $2^n$.