{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $T$ on $(X, \\mu)$ and $S$ on $(Y, \\nu)$ be invertible [[ergodic|AZRPHS478B]] [[measure-preserving transformations|556WBDAHNP]] of [[standard probability spaces|QDDRDW9PXS]], with measurable inverses.\nAssume both have discrete spectrum, in the sense of the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].\nThis means that $L^2$ has an orthonormal basis of [[eigenvectors|0BQAA2BK3Q]] of the [[Koopman operator|37TRG5DBQC]].\nLet $\\Lambda(T) \\subseteq \\C$ be the set of eigenvalues of the Koopman operator $U_T$ of $T$.\n\n1. $T$ and $S$ are isomorphic if and only if $\\Lambda(T) = \\Lambda(S)$.\n2. $\\Lambda(T)$ is a countable subgroup of the unit circle. Moreover, $T$ is isomorphic to a rotation $x \\mapsto x + a$ of a compact metrizable abelian group $K$ with its [[Haar measure|VY0HXD3GC4]], where $\\set{na : n \\in \\Z}$ is dense in $K$.\n\n## Remarks\n\nHere isomorphic means conjugate by a measure-preserving bijection defined up to null sets.\nFor part 1, ergodicity makes every eigenvalue simple and every eigenfunction of constant modulus, so a product of eigenfunctions is again an eigenfunction and $\\Lambda(T)$ is a group.\nChoose for each $\\lambda \\in \\Lambda(T)$ an eigenfunction $f_\\lambda$ of modulus $1$; the products $f_\\lambda f_{\\lambda'}$ and $f_{\\lambda \\lambda'}$ differ by constants.\nThe proof adjusts these constants, which is possible because the circle group is divisible.\nIt then matches the eigenfunctions of $T$ and $S$ to build a unitary map that preserves products, and such a map comes from an isomorphism of the spaces.\nFor part 2, $K$ is the Pontryagin dual of $\\Lambda(T)$ with the discrete topology, and $a$ is the inclusion character.\nExample: for an irrational $\\alpha$, the rotation $x \\mapsto x + \\alpha \\bmod 1$ has $\\Lambda = \\set{e^{2\\pi i k \\alpha} : k \\in \\Z}$.\nSo two irrational rotations by $\\alpha$ and $\\beta$ are isomorphic exactly when $\\beta = \\pm \\alpha \\bmod 1$.\nWithout the discrete spectrum hypothesis the eigenvalues do not determine the system: all weakly mixing systems have $\\Lambda = \\set{1}$.\n\n## Explanations\n\n### L1 human\n\nSome systems behave like a collection of clocks running at fixed frequencies, such as rotations of a circle or a torus.\nFor them, the list of frequencies tells everything: two such systems are the same exactly when they have the same frequencies.\nEach of them is in fact a rotation of a compact group.\n\n### L2 human\n\nThis is the first complete classification theorem in ergodic theory, obtained by Halmos and von Neumann in 1942.\nIt shows the power of spectral methods in the class where the spectrum is discrete, and it is the model case for the [[spectral theory of dynamical systems|Z7GZ6SPFA5]].\nFor systems with continuous spectrum spectral invariants are far from complete, which motivated the search for other invariants such as the Kolmogorov–Sinai entropy.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "Z7GZ6SPFA5"
    ],
    "kind": "statement",
    "msc": [
      "37A30",
      "37A35",
      "37A05"
    ],
    "name": "Halmos–von Neumann theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Measure-preserving_dynamical_system"
      }
    ],
    "tag": "AS18AWFXSB",
    "uses": [
      "QDDRDW9PXS",
      "556WBDAHNP",
      "AZRPHS478B",
      "37TRG5DBQC",
      "Z7GZ6SPFA5",
      "0BQAA2BK3Q",
      "VY0HXD3GC4"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/AS18AWFXSB/",
  "links": [
    {
      "field": "about",
      "target": "Z7GZ6SPFA5"
    },
    {
      "field": "uses",
      "target": "QDDRDW9PXS"
    },
    {
      "field": "uses",
      "target": "556WBDAHNP"
    },
    {
      "field": "uses",
      "target": "AZRPHS478B"
    },
    {
      "field": "uses",
      "target": "37TRG5DBQC"
    },
    {
      "field": "uses",
      "target": "Z7GZ6SPFA5"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "VY0HXD3GC4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AS18AWFXSB.md",
  "msc": [
    "37A30",
    "37A35",
    "37A05"
  ],
  "name": "Halmos–von Neumann theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AS1/AS18AWFXSB.md",
  "status": null,
  "tag": "AS18AWFXSB",
  "uri": "m2://AS18AWFXSB",
  "url": "https://tamnd.github.io/math2/wiki/halmos-von-neumann-theorem/",
  "writing": {
    "tag": "AS18AWFXSB",
    "kind": "statement",
    "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
      }
    ]
  }
}