{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nFor $r \\in [0, 4]$ let $f_r(x) = r x (1 - x)$ be the [[logistic map|QN9X9BNKTX]] on $[0, 1]$.\nLet $A$ be the set of parameters $r \\in [0, 4]$ for which $f_r$ has an [[invariant|B232EZ1FMB]] probability measure that is [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].\nThen $A$ has positive Lebesgue measure.\nMoreover, $4$ is a one-sided Lebesgue density point of $A$:\n$$\\lim_{\\varepsilon \\to 0^+} \\frac{\\lambda(A \\cap [4 - \\varepsilon, 4])}{\\varepsilon} = 1,$$\nwhere $\\lambda$ is Lebesgue measure on $\\R$.\n\n## Remarks\n\nMichael Jakobson proved the theorem in 1981.\nThe proof is a parameter exclusion argument.\nNear $r = 4$, one follows the orbit of the critical point $1/2$ and removes the parameters for which this orbit comes too close to $1/2$ too early.\nFor the remaining parameters the derivative along the critical orbit grows exponentially, and this growth gives the absolutely continuous invariant measure; the removed set is shown to be small.\nBenedicks and Carleson later gave another proof of this kind, which they extended to the Hénon maps.\n\nExample: $r = 4$ itself is in $A$: the measure with density $\\frac{1}{\\pi \\sqrt{x (1 - x)}}$ on $(0, 1)$ is invariant for $f_4$.\n\nFor parameters in $A$ the measure is ergodic and has a positive [[Lyapunov exponent|NJ1FJSNK6A]], so the dynamics is chaotic in a measurable sense.\nThis is in contrast with parameters where $f_r$ has an attracting periodic orbit; by [[Singer's theorem|Y2DKZH3CCT]] there is at most one, and then almost every orbit converges to it.\nSuch parameters form an open and dense subset of $[0, 4]$, but by Jakobson's theorem they do not have full measure.\nLyubich showed in 2002 that almost every parameter $r$ is of one of these two kinds.\n\n## Explanations\n\n### L1 human\n\nIn the logistic family, some parameters give a stable cycle and others give chaos.\nStable cycles are robust: a small change of the parameter keeps the cycle, so these parameters form open intervals that are everywhere in the parameter line.\nOne might expect chaos to be rare then, like a set of measure zero.\nJakobson's theorem says no: chaotic parameters, with a smooth invariant density, fill a set of positive length, and near $r = 4$ almost all parameters are of this kind.\n\n### L2 human\n\nThe theorem shows that the chaotic behaviour seen at $r = 4$ persists, in a measure-theoretic sense, for many nearby parameters.\nThis holds even though the topological picture changes at a dense set of parameters.\nIt became a model for the study of nonuniformly hyperbolic systems with critical points.\nThe tool of following the critical orbit and excluding bad parameters was later used for the Hénon family and other strange attractors.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "QN9X9BNKTX"
    ],
    "kind": "statement",
    "msc": [
      "37E05",
      "37C40",
      "37D25"
    ],
    "name": "Jakobson's theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Logistic_map"
      }
    ],
    "tag": "W49VGF1WDE",
    "uses": [
      "QN9X9BNKTX",
      "P0P21KDV6E",
      "B232EZ1FMB",
      "2Z2QVWZ262",
      "NJ1FJSNK6A",
      "Y2DKZH3CCT"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/W49VGF1WDE/",
  "links": [
    {
      "field": "about",
      "target": "QN9X9BNKTX"
    },
    {
      "field": "uses",
      "target": "QN9X9BNKTX"
    },
    {
      "field": "uses",
      "target": "P0P21KDV6E"
    },
    {
      "field": "uses",
      "target": "B232EZ1FMB"
    },
    {
      "field": "uses",
      "target": "2Z2QVWZ262"
    },
    {
      "field": "uses",
      "target": "NJ1FJSNK6A"
    },
    {
      "field": "uses",
      "target": "Y2DKZH3CCT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/W49VGF1WDE.md",
  "msc": [
    "37E05",
    "37C40",
    "37D25"
  ],
  "name": "Jakobson's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/W49/W49VGF1WDE.md",
  "status": null,
  "tag": "W49VGF1WDE",
  "uri": "m2://W49VGF1WDE",
  "url": "https://tamnd.github.io/math2/wiki/jakobsons-theorem/",
  "writing": {
    "tag": "W49VGF1WDE",
    "kind": "statement",
    "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
      }
    ]
  }
}