{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set, let $f \\colon X \\to X$ be a [[function|0EPAT51DXQ]], and let $\\lambda \\in \\C$.\n*Schröder's equation* for $f$ and $\\lambda$ is the [[functional equation|SRAN4EKWS0]]\n$$\\Psi(f(x)) = \\lambda \\, \\Psi(x) \\quad \\text{for all } x \\in X,$$\nin which the unknown is a function $\\Psi \\colon X \\to \\C$.\nIn the main case, $X$ is a neighbourhood of a [[fixed point|R6QVS6P21N]] $0$ of a holomorphic map $f$, and $\\lambda = f'(0)$.\nThen one asks for a holomorphic $\\Psi$ with $\\Psi(0) = 0$ and $\\Psi'(0) = 1$.\n\n## Remarks\n\nExample: $f(x) = x^2 + 2x$ on $X = (-1, \\infty)$, with $\\lambda = 2$.\nThe fixed point is $0$, and $f'(0) = 2$.\nThe function $\\Psi(x) = \\log(1 + x)$ is a solution with $\\Psi(0) = 0$ and $\\Psi'(0) = 1$, since $\\Psi(f(x)) = \\log\\big((1+x)^2\\big) = 2 \\log(1 + x)$.\n\nNon-example: for the same $f$, the function $\\Psi(x) = x$ is not a solution, since $f(x) = x^2 + 2x \\ne 2x$ for $x \\ne 0$.\n\nIf $\\Psi$ is invertible, the equation says $f = \\Psi^{-1} \\circ (\\lambda \\Psi)$, so $f^n = \\Psi^{-1} \\circ (\\lambda^n \\Psi)$ for every $n \\ge 0$.\nThe function $\\Psi$ is called a Schröder function of $f$.\nIf $\\Psi$ is a solution, so is $c \\Psi$ for every constant $c$; the condition $\\Psi'(0) = 1$ fixes this freedom.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VTJV7JVJEA",
    "kind": "definition",
    "msc": [
      "39B12",
      "30D05",
      "39B32"
    ],
    "name": "Schröder's equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Schr%C3%B6der%27s_equation"
      }
    ],
    "tag": "K4C77FGSW6",
    "uses": [
      "0EPAT51DXQ",
      "R6QVS6P21N",
      "SRAN4EKWS0"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/K4C77FGSW6/",
  "links": [
    {
      "field": "defines",
      "target": "VTJV7JVJEA"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    },
    {
      "field": "uses",
      "target": "R6QVS6P21N"
    },
    {
      "field": "uses",
      "target": "SRAN4EKWS0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/K4C77FGSW6.md",
  "msc": [
    "39B12",
    "30D05",
    "39B32"
  ],
  "name": "Schröder's equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K4C/K4C77FGSW6.md",
  "status": null,
  "tag": "K4C77FGSW6",
  "uri": "m2://K4C77FGSW6",
  "url": "https://tamnd.github.io/math2/wiki/schroders-equation-definition/",
  "writing": {
    "tag": "K4C77FGSW6",
    "kind": "definition",
    "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
      }
    ]
  }
}