{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "19JFF3GY9F"
    },
    {
      "field": "uses",
      "target": "5N5SDY3K53"
    },
    {
      "field": "uses",
      "target": "6HVZ30TS5F"
    },
    {
      "field": "uses",
      "target": "Q7GKAMT3P9"
    },
    {
      "field": "illustrates",
      "target": "XG4D9FF51T"
    }
  ],
  "body": "\n## Example\n\nLet $\\alpha$ be an [[irrational number|0DZXAXEJQ9]] and let $\\mathbb{T} = \\R/\\Z$ be the circle, identified with $[0, 1)$.\nThe *irrational rotation* by $\\alpha$ is\n$$R_\\alpha \\colon \\mathbb{T} \\to \\mathbb{T}, \\qquad R_\\alpha(x) = x + \\alpha \\bmod 1.$$\n\nIt is a [[measure-preserving transformation|556WBDAHNP]] for [[Lebesgue measure|2Z2QVWZ262]] $\\lambda$.\nThe preimage of an arc is an arc of the same length, and arcs generate the Borel sets.\n\nIt is [[ergodic|AZRPHS478B]]: let $f \\in L^2(\\lambda)$ with $f \\circ R_\\alpha = f$, with Fourier coefficients $c_k$.\nThen $c_k e^{2\\pi i k \\alpha} = c_k$ for all $k \\in \\Z$.\nSince $k \\alpha \\notin \\Z$ for $k \\ne 0$, one gets $c_k = 0$ for $k \\ne 0$, so $f$ is constant almost everywhere.\n\nIt is even [[uniquely ergodic|NZZE9KCAD8]], and every orbit is dense in $\\mathbb{T}$.\nBy [[Weyl's equidistribution theorem|NCCGNTBEYV]] every orbit $(x + n\\alpha \\bmod 1)_{n \\ge 0}$ is equidistributed.\n\nIt is not [[weakly mixing|2MGZFRH5PX]]: $e^{2\\pi i x}$ is a nonconstant eigenfunction of the Koopman operator, with eigenvalue $e^{2\\pi i \\alpha}$.\nIts Kolmogorov–Sinai entropy is $0$.\nIf $\\alpha = p/q$ is rational in lowest terms, then $R_\\alpha^q$ is the identity and $R_\\alpha$ is not ergodic.\n\n## Explanations\n\n### L1 human\n\nTurn a wheel by a fixed angle again and again, where the angle is an irrational fraction of a full turn.\nThe marked point never comes back exactly to where it started, and over time it visits every part of the rim equally often.\nThis makes the irrational rotation the simplest ergodic system.\nIt is orderly rather than chaotic: nearby points stay at the same distance forever, and the system has zero entropy.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "556WBDAHNP",
      "AZRPHS478B",
      "NZZE9KCAD8"
    ],
    "kind": "example",
    "msc": [
      "37E10",
      "37A05",
      "37A25"
    ],
    "name": "Irrational rotation",
    "same_as": {
      "wikidata": "Q6073537",
      "wikipedia": "Irrational rotation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Irrational_rotation"
      }
    ],
    "tag": "2Q1J7RZ4TK",
    "uses": [
      "0DZXAXEJQ9",
      "2Z2QVWZ262",
      "NCCGNTBEYV",
      "2MGZFRH5PX"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/2Q1J7RZ4TK/",
  "links": [
    {
      "field": "instance_of",
      "target": "556WBDAHNP"
    },
    {
      "field": "instance_of",
      "target": "AZRPHS478B"
    },
    {
      "field": "instance_of",
      "target": "NZZE9KCAD8"
    },
    {
      "field": "uses",
      "target": "0DZXAXEJQ9"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "NCCGNTBEYV"
    },
    {
      "field": "uses",
      "target": "2MGZFRH5PX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2Q1J7RZ4TK.md",
  "msc": [
    "37E10",
    "37A05",
    "37A25"
  ],
  "name": "Irrational rotation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2Q1/2Q1J7RZ4TK.md",
  "status": null,
  "tag": "2Q1J7RZ4TK",
  "uri": "m2://2Q1J7RZ4TK",
  "url": "https://tamnd.github.io/math2/wiki/irrational-rotation/",
  "writing": {
    "tag": "2Q1J7RZ4TK",
    "kind": "example",
    "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
      }
    ]
  }
}