{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5GNSQRHSSC"
    },
    {
      "field": "uses",
      "target": "H27ZC95KN5"
    },
    {
      "field": "uses",
      "target": "X1DB08QJHK"
    },
    {
      "field": "specializes",
      "target": "E7890PHMGP"
    },
    {
      "field": "defines",
      "target": "FE8EYTGMRS"
    }
  ],
  "body": "\n## Summary\n\nAn iterative root of order $n$ of a map $f$ is a map $g$ whose $n$-th [[iterate|AYVTR2ES5B]] is $f$, that is, $g \\circ g \\circ \\cdots \\circ g = f$ with $n$ factors.\nIt is the analogue for composition of the $n$-th root of a number.\nIterative roots may fail to exist, and when they exist there are often many of them.\nThe search for them is one of the classical problems of iteration theory.\n\n## Explanations\n\n### L1 human\n\nThink of $f$ as a step of a process, and ask for a smaller step $g$ such that $n$ small steps make one big step.\nFor $f(x) = x + 3$, the map $g(x) = x + 1$ is an iterative root of order $3$, since three shifts by $1$ make a shift by $3$.\nFor $f(x) = 8x$, the map $g(x) = 2x$ is an iterative root of order $3$.\nSome maps have no nice root: no continuous map $g$ of the real line satisfies $g(g(x)) = -x$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "39B12",
      "26A18"
    ],
    "name": "Iterative root",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Functional_square_root"
      }
    ],
    "tag": "27C6J3YNZY"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/27C6J3YNZY/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/27C6J3YNZY.md",
  "msc": [
    "39B12",
    "26A18"
  ],
  "name": "Iterative root",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/27C/27C6J3YNZY.md",
  "status": null,
  "tag": "27C6J3YNZY",
  "uri": "m2://27C6J3YNZY",
  "url": "https://tamnd.github.io/math2/wiki/iterative-root/",
  "writing": {
    "tag": "27C6J3YNZY",
    "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
      }
    ]
  }
}