{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *topological dynamical system* is a pair $(X, T)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $T \\colon X \\to X$ is a [[continuous map|0VGFHTM75Y]].\nMore generally, it is a triple $(G, X, \\alpha)$ where $G$ is a [[topological group|92700QAB5Y]] and $\\alpha \\colon G \\times X \\to X$ is a continuous [[group action|0WCY42EV9V]].\n*Topological dynamics* is the study of the properties of such systems that are preserved by topological conjugacy, that is, by homeomorphisms $h \\colon X \\to Y$ with $h \\circ T = S \\circ h$.\n\n## Remarks\n\nIn most of the theory $X$ is a compact metric space, and $T$ is a homeomorphism or $G$ is $\\Z$ or $\\R$.\nA map $T$ gives an action of the monoid $\\N$ by $n \\cdot x = T^n(x)$; a homeomorphism gives an action of $\\Z$; a continuous flow gives an action of $\\R$.\nExample: the circle $X = \\R / \\Z$ with $T(x) = x + \\alpha$ is a topological dynamical system, since $T$ is continuous.\nNon-example: $X = [0, 1]$ with $T(x) = 0$ for $x \u003c 1/2$ and $T(x) = 1$ for $x \\ge 1/2$ is not one, because $T$ is not continuous at $1/2$.\n\n## Explanations\n\n### L1 human\n\nA topological dynamical system is a space together with a continuous rule of motion.\nTopological dynamics studies the features of the motion that survive a continuous change of coordinates.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "5X6E97EMFR",
    "kind": "definition",
    "msc": [
      "37B02",
      "37B05"
    ],
    "name": "Topological dynamics",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Topological_dynamics"
      }
    ],
    "tag": "1Q901WTV5X",
    "uses": [
      "0CQHFPGJ45",
      "0VGFHTM75Y",
      "92700QAB5Y",
      "0WCY42EV9V"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/1Q901WTV5X/",
  "links": [
    {
      "field": "defines",
      "target": "5X6E97EMFR"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0VGFHTM75Y"
    },
    {
      "field": "uses",
      "target": "92700QAB5Y"
    },
    {
      "field": "uses",
      "target": "0WCY42EV9V"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1Q901WTV5X.md",
  "msc": [
    "37B02",
    "37B05"
  ],
  "name": "Topological dynamics",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1Q9/1Q901WTV5X.md",
  "status": null,
  "tag": "1Q901WTV5X",
  "uri": "m2://1Q901WTV5X",
  "url": "https://tamnd.github.io/math2/wiki/topological-dynamics-definition/",
  "writing": {
    "tag": "1Q901WTV5X",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}