{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \\colon X \\to X$ be a [[continuous map|0X3WGX95FE]], with [[iterates|AYVTR2ES5B]] $f^n$.\nFor $x \\in X$ and $A \\subseteq X$ nonempty, let $d(x, A) = \\inf_{a \\in A} d(x, a)$.\nA nonempty [[compact|0TVJZYB6SH]] set $A \\subseteq X$ is an *attractor* of $f$ if:\n\n1. $A$ is invariant: $f(A) = A$;\n2. $A$ attracts an open neighbourhood: there is an [[open set|02BH3MCBQB]] $U \\supseteq A$ such that $d(f^n(x), A) \\to 0$ as $n \\to \\infty$ for every $x \\in U$;\n3. $A$ is minimal: no nonempty compact proper subset of $A$ satisfies clauses 1 and 2.\n\nThe *basin of attraction* of $A$ is the set of all $x \\in X$ with $d(f^n(x), A) \\to 0$.\nFor a flow $(\\varphi_t)$ the definition is the same, with $\\varphi_t(A) = A$ for all $t$ and $t \\to \\infty$ in place of $n \\to \\infty$.\n\n## Remarks\n\nA set that satisfies clauses 1 and 2 is often called an attracting set.\nThe literature uses several other versions.\nSome replace clause 3 by the existence of a dense orbit in $A$.\nSome ask for a neighbourhood $U$ with $f(\\overline{U}) \\subseteq U$ and $A = \\bigcap_{n \\ge 0} f^n(U)$.\nMilnor's version replaces the open neighbourhood in clause 2 by a basin of positive measure.\n\nExample: $f(x) = x/2$ on $\\R$ and $A = \\set{0}$.\nClause 1 holds since $f(0) = 0$; clause 2 holds with $U = \\R$, since $d(f^n(x), A) = \\abs{x}/2^n \\to 0$; clause 3 holds since $A$ has no nonempty proper subset.\n\nNon-example: let $f$ be the time-$1$ map of the flow of $u' = u - u^3$ on $\\R$, and let $A = [-1, 1]$.\nEvery solution tends to $-1$, $0$ or $1$, so $A$ is invariant and attracts all of $\\R$.\nBut the proper subset $\\set{1}$ also satisfies clauses 1 and 2, with $U = (0, \\infty)$, so clause 3 fails for $A$.\nThe attractors of this map are $\\set{-1}$ and $\\set{1}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "7FWX24X5MW",
    "kind": "definition",
    "msc": [
      "37C70",
      "37G35",
      "37B25"
    ],
    "name": "Attractor",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Attractor"
      }
    ],
    "tag": "EZ24HDESPB",
    "uses": [
      "0VKD1TP7F3",
      "0X3WGX95FE",
      "0TVJZYB6SH",
      "02BH3MCBQB",
      "AYVTR2ES5B"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/EZ24HDESPB/",
  "links": [
    {
      "field": "defines",
      "target": "7FWX24X5MW"
    },
    {
      "field": "uses",
      "target": "0VKD1TP7F3"
    },
    {
      "field": "uses",
      "target": "0X3WGX95FE"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "AYVTR2ES5B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EZ24HDESPB.md",
  "msc": [
    "37C70",
    "37G35",
    "37B25"
  ],
  "name": "Attractor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EZ2/EZ24HDESPB.md",
  "status": null,
  "tag": "EZ24HDESPB",
  "uri": "m2://EZ24HDESPB",
  "url": "https://tamnd.github.io/math2/wiki/attractor-definition/",
  "writing": {
    "tag": "EZ24HDESPB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}