{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7E2ZX7N4VS"
    },
    {
      "field": "uses",
      "target": "9GWGW4YRD9"
    },
    {
      "field": "uses",
      "target": "BZ0DDT1MXS"
    },
    {
      "field": "uses",
      "target": "G02NBNN6CJ"
    },
    {
      "field": "uses",
      "target": "JTERX19M7X"
    },
    {
      "field": "uses",
      "target": "K4C77FGSW6"
    },
    {
      "field": "uses",
      "target": "MS8G1FD8FD"
    },
    {
      "field": "uses",
      "target": "QN9X9BNKTX"
    },
    {
      "field": "uses",
      "target": "S5HVHEPSMA"
    },
    {
      "field": "uses",
      "target": "SBF5C86E3Z"
    },
    {
      "field": "uses",
      "target": "TE3STTNSCK"
    },
    {
      "field": "uses",
      "target": "YBBDYHBADD"
    },
    {
      "field": "specializes",
      "target": "D62PEKHRV5"
    },
    {
      "field": "about",
      "target": "G02NBNN6CJ"
    },
    {
      "field": "about",
      "target": "JTERX19M7X"
    },
    {
      "field": "about",
      "target": "YBBDYHBADD"
    },
    {
      "field": "defines",
      "target": "W46BP9W06H"
    }
  ],
  "body": "\n## Summary\n\nA fixed point of a map is a point that the map sends to itself.\nIn a dynamical system, a fixed point is a state that never changes, the discrete-time analogue of an [[equilibrium point|0N8NGKDZDZ]].\nMany problems, such as solving an equation $g(x) = 0$, can be rewritten as finding a fixed point.\nTheorems such as the [[Banach fixed point theorem|0VK6VEH345]] guarantee that one exists.\nNear a fixed point, the behaviour of the iterates tells whether nearby states are attracted or repelled.\n\n## Explanations\n\n### L1 human\n\nA fixed point is a point that a map leaves where it is.\nFor $f(x) = x^2$ on the real line, the fixed points are the solutions of $x^2 = x$, namely $0$ and $1$.\nOn a graph, the fixed points are where the graph of $f$ meets the diagonal line $y = x$.\nStir a cup of coffee and let it settle.\nIf the stirring is a continuous map of the disc to itself, at least one point of the coffee ends where it started.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37C25",
      "54H25",
      "37-01"
    ],
    "name": "Fixed point",
    "same_as": {
      "wikidata": "Q217608",
      "wikipedia": "Fixed point (mathematics)"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fixed_point_(mathematics)"
      }
    ],
    "tag": "R6QVS6P21N"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/R6QVS6P21N/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/R6QVS6P21N.md",
  "msc": [
    "37C25",
    "54H25",
    "37-01"
  ],
  "name": "Fixed point",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/R6Q/R6QVS6P21N.md",
  "status": null,
  "tag": "R6QVS6P21N",
  "uri": "m2://R6QVS6P21N",
  "url": "https://tamnd.github.io/math2/wiki/fixed-point/",
  "writing": {
    "tag": "R6QVS6P21N",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}