{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $f \\colon \\C \\to \\C$ be a transcendental [[entire function|0G4F0XB9X2]].\nLet $I(f) = \\set{z \\in \\C : f^n(z) \\to \\infty \\text{ as } n \\to \\infty}$ be its escaping set.\nThen every [[connected component|0JG9W3KA6B]] of $I(f)$ is unbounded.\n\n## Remarks\n\nEremenko proved that $I(f)$ is nonempty and that the [[Julia set|5M4S7P04GR]] of $f$ is the boundary of $I(f)$.\nHe also proved that every component of the closure of $I(f)$ is unbounded.\nRippon and Stallard proved in 2005 that $I(f)$ always has at least one unbounded component.\nA stronger form, that every escaping point can be joined to $\\infty$ by a curve in $I(f)$, is false.\nRottenfußer, Rückert, Rempe and Schleicher gave entire functions whose escaping sets contain no curves at all.\nIn 2021 Martí-Pete, Rempe and Waterman constructed a counterexample to the conjecture itself.\nFor a polynomial of degree at least $2$ the analogous statement holds.\nThere the escaping set is the complement of the filled Julia set $K$, which is compact and has connected complement.\nSo the escaping set is connected and unbounded.\n\n## Explanations\n\n### L1 human\n\nFor a transcendental entire function such as $e^z$, some points run off to infinity under iteration; together they form the escaping set.\nEremenko guessed that this set has no bounded pieces: every piece should reach out to infinity.\nFor many families, such as the exponential maps, the escaping set is indeed made of curves that go to infinity.\nIn general the guess turned out to be false, but the question shaped much of the study of transcendental dynamics.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0G4F0XB9X2"
    ],
    "history": {
      "first_stated": "1989",
      "posed_by": "Alexandre Eremenko"
    },
    "kind": "conjecture",
    "msc": [
      "37F10",
      "30D05"
    ],
    "name": "Eremenko's conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Escaping_set"
      }
    ],
    "tag": "WWSYQJF7TZ",
    "uses": [
      "0G4F0XB9X2",
      "5M4S7P04GR",
      "0JG9W3KA6B"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/WWSYQJF7TZ/",
  "links": [
    {
      "field": "about",
      "target": "0G4F0XB9X2"
    },
    {
      "field": "uses",
      "target": "0G4F0XB9X2"
    },
    {
      "field": "uses",
      "target": "5M4S7P04GR"
    },
    {
      "field": "uses",
      "target": "0JG9W3KA6B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WWSYQJF7TZ.md",
  "msc": [
    "37F10",
    "30D05"
  ],
  "name": "Eremenko's conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WWS/WWSYQJF7TZ.md",
  "status": null,
  "tag": "WWSYQJF7TZ",
  "uri": "m2://WWSYQJF7TZ",
  "url": "https://tamnd.github.io/math2/wiki/eremenkos-conjecture/",
  "writing": {
    "tag": "WWSYQJF7TZ",
    "kind": "conjecture",
    "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
      }
    ]
  }
}