{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "BZ0DDT1MXS"
    }
  ],
  "body": "\n## Summary\n\nPoincaré's functional equation asks for a function $\\Psi$ with $\\Psi(\\lambda z) = f(\\Psi(z))$, where $f$ is a given holomorphic map with a [[fixed point|R6QVS6P21N]] at $0$ and $\\lambda = f'(0)$.\nIt is the inverse form of [[Schröder's equation|VTJV7JVJEA]]: if $\\Phi$ solves Schröder's equation and is invertible, then $\\Psi = \\Phi^{-1}$ solves Poincaré's equation.\nFor a repelling fixed point of an entire map, the solution extends to an entire function on all of $\\C$, the Poincaré function.\nPoincaré functions parametrise the dynamics of $f$ near the repelling point and are used to study Julia sets.\n\n## Explanations\n\n### L1 human\n\nSchröder's equation looks for a coordinate in which the map $f$ becomes multiplication by $\\lambda$.\nPoincaré's equation looks at the same change of coordinates in the other direction: it maps the simple picture, multiplication by $\\lambda$, onto the complicated one.\nFor $f(w) = w^2 + 2w$ and $\\lambda = 2$, the function $\\Psi(z) = e^z - 1$ works, since $f(e^z - 1) = e^{2z} - 1 = \\Psi(2z)$.\nFor the logistic map $f(w) = 4w(1 - w)$ and $\\lambda = 4$, the function $\\Psi(z) = \\sin^2 \\sqrt{z}$ works, by the double-angle formula.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "39B32",
      "30D05",
      "37F10"
    ],
    "name": "Poincaré's functional equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schr%C3%B6der%27s_equation"
      }
    ],
    "specializes": [
      "BCGBQ3HK26"
    ],
    "tag": "3384ZV3PWM"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/3384ZV3PWM/",
  "links": [
    {
      "field": "specializes",
      "target": "BCGBQ3HK26"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/3384ZV3PWM.md",
  "msc": [
    "39B32",
    "30D05",
    "37F10"
  ],
  "name": "Poincaré's functional equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/338/3384ZV3PWM.md",
  "status": null,
  "tag": "3384ZV3PWM",
  "uri": "m2://3384ZV3PWM",
  "url": "https://tamnd.github.io/math2/wiki/poincares-functional-equation/",
  "writing": {
    "tag": "3384ZV3PWM",
    "kind": "concept",
    "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
      }
    ]
  }
}