{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5N5SDY3K53"
    },
    {
      "field": "uses",
      "target": "G02NBNN6CJ"
    },
    {
      "field": "uses",
      "target": "N63Z6EA6YS"
    },
    {
      "field": "uses",
      "target": "Q7GKAMT3P9"
    },
    {
      "field": "uses",
      "target": "QN9X9BNKTX"
    },
    {
      "field": "uses",
      "target": "TMM9SGJMSP"
    },
    {
      "field": "uses",
      "target": "YBBDYHBADD"
    },
    {
      "field": "defines",
      "target": "CTCSAWBSB8"
    }
  ],
  "body": "\n## Summary\n\nTwo maps are topologically conjugate when a [[homeomorphism|0QT00PB5G2]] between their spaces turns one map into the other.\nConjugate systems have the same dynamics up to a continuous change of coordinates: fixed points, periodic orbits, dense orbits and attractors correspond.\nTopological conjugacy is the basic notion of \"same system\" in topological dynamics, and many classification results, such as structural stability, are stated with it.\nIf the homeomorphism is only required to be a continuous surjection, one gets the weaker notion of a [[semiconjugacy|AEXJNRBASV]].\n\n## Explanations\n\n### L1 human\n\nImagine two maps that do the same thing, but one is drawn on a rubber sheet that has been stretched.\nA conjugacy is the stretching that matches them up, step by step.\nFor example, $f(x) = x/2$ and $g(x) = x/8$ on the real line are conjugate through $h(x) = x^3$.\nIndeed, $h(f(x)) = x^3/8 = g(h(x))$, so moving by $f$ and then changing coordinates is the same as changing coordinates and then moving by $g$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "37C15",
      "37B02"
    ],
    "name": "Topological conjugacy",
    "same_as": {
      "wikidata": "Q2443460",
      "wikipedia": "Topological conjugacy"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Topological_conjugacy"
      }
    ],
    "tag": "HNXXAPQKG3"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/HNXXAPQKG3/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/HNXXAPQKG3.md",
  "msc": [
    "37C15",
    "37B02"
  ],
  "name": "Topological conjugacy",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HNX/HNXXAPQKG3.md",
  "status": null,
  "tag": "HNXXAPQKG3",
  "uri": "m2://HNXXAPQKG3",
  "url": "https://tamnd.github.io/math2/wiki/topological-conjugacy/",
  "writing": {
    "tag": "HNXXAPQKG3",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}