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  "body": "\n## Summary\n\nThe Artin–Mazur zeta function of a map packs the numbers of [[periodic points|T75CZ5H47N]] of all periods into one power series.\nFor 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 $f^n$.\nFor many hyperbolic systems it is a rational function, and it then records all periodic orbit counts by finitely many numbers.\nFor the Frobenius map of a variety over a finite field it is the local zeta function of the variety.\n\n## Explanations\n\n### L1 human\n\nCounting periodic points of a map gives a list of numbers $N_1, N_2, N_3, \\dots$, one for each period.\nThe zeta function turns the whole list into a single function of one variable.\nFor 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 $(1 - z)/(1 - 2z)$.\nThe pole at $z = 1/2$ shows that the number of periodic points grows like $2^n$.\n",
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