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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a map.\nAssume that, for every $n \\ge 1$, the set $\\operatorname{Fix}(f^n)$ of [[fixed points|R6QVS6P21N]] of the $n$-th iterate $f^n$ is finite.\nWrite $N_n = \\abs{\\operatorname{Fix}(f^n)}$.\nThe *Artin–Mazur zeta function* of $f$ is the [[formal power series|033WGN00BF]]\n$$\\zeta_f(z) = \\exp\\left( \\sum_{n=1}^{\\infty} \\frac{N_n}{n} z^n \\right).$$\nIt has rational coefficients.\n\n## Remarks\n\nExample: let $f(z) = z^2$ on the unit circle in $\\C$.\nThe fixed points of $f^n$ are the solutions of $z^{2^n} = z$ with $\\abs{z} = 1$, that is, the $(2^n - 1)$-th roots of unity, so $N_n = 2^n - 1$.\nThen\n$$\\zeta_f(z) = \\exp\\left( \\sum_{n \\ge 1} \\frac{(2z)^n}{n} - \\sum_{n \\ge 1} \\frac{z^n}{n} \\right) = \\frac{1 - z}{1 - 2z}.$$\n\nNon-example: the identity map of the circle has infinitely many fixed points, so $N_1$ is not finite and the definition does not apply.\n\nIf $N_n \\le C a^n$ for constants $C$ and $a$, the series has a positive [[radius of convergence|0N288P2TB5]] and $\\zeta_f$ is a holomorphic function near $0$.\nFor the shift on a subshift of finite type with transition matrix $A$, $N_n = \\tr(A^n)$ and $\\zeta_f(z) = 1 / \\det(I - zA)$.\nFor an Axiom A diffeomorphism of a compact manifold, the zeta function is rational (Manning).\nThe function has the product form $\\zeta_f(z) = \\prod_{\\gamma} (1 - z^{\\abs{\\gamma}})^{-1}$, a product over the periodic orbits $\\gamma$ of $f$ with $\\abs{\\gamma}$ the number of points of $\\gamma$.\n",
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