{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a map.\nAssume that the set $\\operatorname{Fix}(f^n)$ of [[fixed points|R6QVS6P21N]] of $f^n$ is finite for every $n \\ge 1$.\nLet $\\varphi \\colon X \\to \\C$ be a function, called the weight.\nThe *Ruelle zeta function* of $(f, \\varphi)$ is the [[formal power series|033WGN00BF]]\n$$\\zeta_{f, \\varphi}(z) = \\exp\\left( \\sum_{n=1}^{\\infty} \\frac{z^n}{n} \\sum_{x \\in \\operatorname{Fix}(f^n)} \\prod_{k=0}^{n-1} \\varphi(f^k x) \\right).$$\n\n## Remarks\n\nExample: let $f$ be the shift on the full shift $\\set{0, 1}^{\\N}$ and let $\\varphi = c$ be a constant.\nThe shift $f^n$ has $2^n$ fixed points, each of weight $c^n$, so $\\zeta_{f, \\varphi}(z) = \\exp \\sum_{n \\ge 1} (2cz)^n / n = 1/(1 - 2cz)$.\n\nNon-example: for the identity map of the circle the set $\\operatorname{Fix}(f)$ is infinite, so the inner sum is not defined.\n\nWith $\\varphi = 1$ one gets the [[Artin–Mazur zeta function|2A5Z21CE4P]] of $f$.\nMore generally the weight may take values in $d \\times d$ matrices, and then the product is replaced by the trace of the ordered product $\\varphi(f^{n-1} x) \\cdots \\varphi(x)$.\nIf $f$ is the shift on a [[subshift of finite type|624XDXX8FN]] with transition matrix $A$ and $\\varphi(x) = w(x_0, x_1)$ depends only on the first two symbols, then $\\zeta_{f, \\varphi}(z) = 1/\\det(I - z B)$, where $B_{ij} = A_{ij} w(i, j)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ENYXKW76AA",
    "kind": "definition",
    "msc": [
      "37C30",
      "11M41"
    ],
    "name": "Ruelle zeta function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ruelle_zeta_function"
      }
    ],
    "tag": "MS8G1FD8FD",
    "uses": [
      "2A5Z21CE4P",
      "R6QVS6P21N",
      "033WGN00BF",
      "624XDXX8FN"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MS8G1FD8FD/",
  "links": [
    {
      "field": "defines",
      "target": "ENYXKW76AA"
    },
    {
      "field": "uses",
      "target": "2A5Z21CE4P"
    },
    {
      "field": "uses",
      "target": "R6QVS6P21N"
    },
    {
      "field": "uses",
      "target": "033WGN00BF"
    },
    {
      "field": "uses",
      "target": "624XDXX8FN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MS8G1FD8FD.md",
  "msc": [
    "37C30",
    "11M41"
  ],
  "name": "Ruelle zeta function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MS8/MS8G1FD8FD.md",
  "status": null,
  "tag": "MS8G1FD8FD",
  "uri": "m2://MS8G1FD8FD",
  "url": "https://tamnd.github.io/math2/wiki/ruelle-zeta-function-definition/",
  "writing": {
    "tag": "MS8G1FD8FD",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}