{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\mathcal{A}, \\mu)$ be a [[probability space|0NFPJVJJHX]], and let $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]].\nA set $A \\in \\mathcal{A}$ is *$T$-invariant* if $T^{-1}(A) = A$.\nThe map $T$ is *ergodic* if every $T$-invariant set $A \\in \\mathcal{A}$ satisfies $\\mu(A) = 0$ or $\\mu(A) = 1$.\n\n## Remarks\n\nEquivalent forms: every measurable $f \\colon X \\to \\R$ with $f \\circ T = f$ almost everywhere is constant almost everywhere.\nOne can also require $\\mu(T^{-1}(A) \\mathbin{\\triangle} A) = 0$ instead of $T^{-1}(A) = A$; this gives the same class of maps.\nExample: the rotation $T(x) = x + \\alpha \\bmod 1$ on $[0,1)$ with Lebesgue measure and $\\alpha$ irrational.\nIf $A$ is invariant, the Fourier coefficients of $f = 1_A$ satisfy $\\hat f(k) e^{2 \\pi i k \\alpha} = \\hat f(k)$.\nSince $k \\alpha \\notin \\Z$ for $k \\ne 0$, all $\\hat f(k)$ with $k \\ne 0$ vanish, so $f$ is constant almost everywhere and $\\mu(A) \\in \\set{0, 1}$.\nNon-example: for $\\alpha = 1/2$ the set $A = [0, 1/4) \\cup [1/2, 3/4)$ is invariant with $\\mu(A) = 1/2$.\nThe identity map is not ergodic on $[0,1]$ either, since every measurable set is invariant.\nThe definition extends to $\\sigma$-finite measures: then an invariant set must be null or have null complement.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "AZRPHS478B",
    "kind": "definition",
    "msc": [
      "37A25",
      "28D05"
    ],
    "name": "Ergodic transformation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ergodicity"
      }
    ],
    "tag": "H3SZ6VAJ6F",
    "uses": [
      "0NFPJVJJHX",
      "556WBDAHNP"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/H3SZ6VAJ6F/",
  "links": [
    {
      "field": "defines",
      "target": "AZRPHS478B"
    },
    {
      "field": "uses",
      "target": "0NFPJVJJHX"
    },
    {
      "field": "uses",
      "target": "556WBDAHNP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/H3SZ6VAJ6F.md",
  "msc": [
    "37A25",
    "28D05"
  ],
  "name": "Ergodic transformation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H3S/H3SZ6VAJ6F.md",
  "status": null,
  "tag": "H3SZ6VAJ6F",
  "uri": "m2://H3SZ6VAJ6F",
  "url": "https://math2.tamnd.com/wiki/ergodic-transformation-definition/",
  "writing": {
    "tag": "H3SZ6VAJ6F",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}