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  "body": "\n## Summary\n\nThe Ruelle zeta function of a map is a weighted count of its [[periodic points|T75CZ5H47N]].\nEach fixed point of $f^n$ is counted with the product of a weight function along its orbit.\nWith the weight $1$ it is the [[Artin–Mazur zeta function|2A5Z21CE4P]].\nIt is a basic tool of the thermodynamic formalism, where its poles and zeros are related to the eigenvalues of [[transfer operators|CNZV00JQRC]].\nRuelle also defined a version for flows, with a product over closed orbits.\nFor geodesic flows of hyperbolic surfaces, this version is close to the Selberg zeta function.\n\n## Explanations\n\n### L1 human\n\nInstead of just counting periodic points, one can give each periodic orbit a weight, for example a probability or a stretching factor.\nThe Ruelle zeta function packs these weighted counts for all periods into one function.\nIts singularities then tell how fast the weighted counts grow, which in turn gives the pressure and the rate of mixing of the system.\n",
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