{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AS18AWFXSB"
    },
    {
      "field": "uses",
      "target": "AS18AWFXSB"
    },
    {
      "field": "defines",
      "target": "PV2DGD23AY"
    }
  ],
  "body": "\n## Summary\n\nThe spectral theory of dynamical systems studies a [[measure-preserving transformation|556WBDAHNP]] through the spectrum of its [[Koopman operator|37TRG5DBQC]] on $L^2$.\nTwo systems with unitarily equivalent Koopman operators are called spectrally isomorphic, and properties preserved by this relation are spectral invariants.\nErgodicity, weak mixing, mixing and discrete spectrum are spectral invariants.\nThe theory is complete for ergodic systems with discrete spectrum, by the [[Halmos–von Neumann theorem|AS18AWFXSB]].\nIn general, spectral data do not determine a system up to isomorphism.\n\n## Explanations\n\n### L1 human\n\nThe idea goes back to Koopman and von Neumann, who described classical mechanics by a linear operator on a Hilbert space, as in quantum mechanics.\nOne then asks which frequencies the system has: an eigenfunction with eigenvalue $e^{2\\pi i \\theta}$ is a measurement that turns like a clock at frequency $\\theta$.\nA rotation of the circle has many such clocks; a strongly chaotic system has none besides the constants.\nSpectral data are easy to compute but coarse: the [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] separates systems that the spectrum cannot.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37A30",
      "37A35",
      "37A05"
    ],
    "name": "Spectral theory of dynamical systems",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics"
      }
    ],
    "tag": "Z7GZ6SPFA5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Z7GZ6SPFA5/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/Z7GZ6SPFA5.md",
  "msc": [
    "37A30",
    "37A35",
    "37A05"
  ],
  "name": "Spectral theory of dynamical systems",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z7G/Z7GZ6SPFA5.md",
  "status": null,
  "tag": "Z7GZ6SPFA5",
  "uri": "m2://Z7GZ6SPFA5",
  "url": "https://tamnd.github.io/math2/wiki/spectral-theory-of-dynamical-systems/",
  "writing": {
    "tag": "Z7GZ6SPFA5",
    "kind": "concept",
    "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
      }
    ]
  }
}