{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "8PX1F72APY"
    },
    {
      "field": "illustrates",
      "target": "N63Z6EA6YS"
    }
  ],
  "body": "\n## Summary\n\nSymbolic dynamics studies systems whose states are infinite sequences of symbols from a finite alphabet.\nTheir time evolution is the shift of a sequence by one place.\nThese systems are the [[shift spaces|A1XE59ZHNM]].\nIt is also a method: one codes an orbit of a general system by the pieces of a finite partition that it visits.\nThrough such codings, questions about smooth or hyperbolic systems become combinatorial questions about sequences, as for [[subshifts of finite type|624XDXX8FN]].\n\n## Explanations\n\n### L1 human\n\nDivide the space of a system into a few labelled regions, say $0$ and $1$.\nInstead of following a point exactly, write down the label of the region it visits at each time.\nThe orbit becomes an infinite word such as $0110100\\ldots$, and moving forward in time just deletes the first letter.\nFor the doubling map $x \\mapsto 2x \\bmod 1$, with the regions $[0, 1/2)$ and $[1/2, 1)$, the word of $x$ is its binary expansion.\nSymbolic dynamics studies the sets of words that arise this way.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37B10"
    ],
    "name": "Symbolic dynamics",
    "same_as": {
      "wikidata": "Q1350838",
      "wikipedia": "Symbolic dynamics"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Symbolic_dynamics"
      }
    ],
    "tag": "EWFWTSVY0B"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/EWFWTSVY0B/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/EWFWTSVY0B.md",
  "msc": [
    "37B10"
  ],
  "name": "Symbolic dynamics",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EWF/EWFWTSVY0B.md",
  "status": null,
  "tag": "EWFWTSVY0B",
  "uri": "m2://EWFWTSVY0B",
  "url": "https://tamnd.github.io/math2/wiki/symbolic-dynamics/",
  "writing": {
    "tag": "EWFWTSVY0B",
    "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
      }
    ]
  }
}