{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P$ be a monic [[polynomial|1G0B73WPX8]] of degree $d \\ge 2$ whose filled Julia set $K$ is connected.\nLet $\\overline{\\mathbb{D}}$ be the closed unit disc.\nBy [[Böttcher's theorem|YBBDYHBADD]] there is a unique [[conformal map|0YYT36R4YD]] $\\varphi \\colon \\C \\setminus K \\to \\C \\setminus \\overline{\\mathbb{D}}$ with $\\varphi(z)/z \\to 1$ as $z \\to \\infty$.\nIt satisfies $\\varphi(P(z)) = \\varphi(z)^d$.\nFor $\\theta \\in \\R/\\Z$, the *external ray of angle* $\\theta$ is\n$$R_\\theta = \\varphi^{-1}\\big(\\set{r e^{2\\pi i \\theta} : r \u003e 1}\\big).$$\nThe ray $R_\\theta$ *lands* at $z_0 \\in \\partial K$ if $\\varphi^{-1}(r e^{2\\pi i \\theta}) \\to z_0$ as $r \\to 1^+$.\n\n## Remarks\n\nThe relation $\\varphi \\circ P = \\varphi^d$ gives $P(R_\\theta) = R_{d\\theta}$.\nExample: for $P(z) = z^2$, $K$ is the closed unit disc and $\\varphi$ is the identity.\nSo $R_\\theta$ is the half-line $\\set{r e^{2\\pi i \\theta} : r \u003e 1}$, and it lands at $e^{2\\pi i \\theta}$.\nNon-example: for $P(z) = z^2$, the spiral $\\set{r e^{2\\pi i (\\theta + \\log r)} : r \u003e 1}$ is not an external ray, since $\\varphi$ maps it to a curve whose angle varies.\nDouady and Hubbard showed that every ray with a rational angle lands, at a periodic or preperiodic point of the [[Julia set|5M4S7P04GR]].\nFor the [[quadratic family|WDZFV2ACXR]], external rays of the Mandelbrot set are defined in the same way, with the conformal map from its complement onto $\\C \\setminus \\overline{\\mathbb{D}}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CNX4KVMDJQ",
    "kind": "definition",
    "msc": [
      "37F20",
      "37F10"
    ],
    "name": "External ray",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/External_ray"
      }
    ],
    "tag": "47DYWZTQMC",
    "uses": [
      "YBBDYHBADD",
      "WDZFV2ACXR",
      "5M4S7P04GR",
      "1G0B73WPX8",
      "0YYT36R4YD"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/47DYWZTQMC/",
  "links": [
    {
      "field": "defines",
      "target": "CNX4KVMDJQ"
    },
    {
      "field": "uses",
      "target": "YBBDYHBADD"
    },
    {
      "field": "uses",
      "target": "WDZFV2ACXR"
    },
    {
      "field": "uses",
      "target": "5M4S7P04GR"
    },
    {
      "field": "uses",
      "target": "1G0B73WPX8"
    },
    {
      "field": "uses",
      "target": "0YYT36R4YD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/47DYWZTQMC.md",
  "msc": [
    "37F20",
    "37F10"
  ],
  "name": "External ray",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/47D/47DYWZTQMC.md",
  "status": null,
  "tag": "47DYWZTQMC",
  "uri": "m2://47DYWZTQMC",
  "url": "https://tamnd.github.io/math2/wiki/external-ray-definition/",
  "writing": {
    "tag": "47DYWZTQMC",
    "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
      }
    ]
  }
}