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  "body": "\n## Definition\n\nLet $f \\colon \\hat{\\C} \\to \\hat{\\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$ on the [[Riemann sphere|0J78KBPFZB]] $\\hat{\\C}$.\nLet $f^n$ be its $n$-th [[iterate|AYVTR2ES5B]].\nA point $z \\in \\hat{\\C}$ belongs to the *Fatou set* $F(f)$ if the following holds.\nThere is an [[open|02BH3MCBQB]] neighbourhood $U$ of $z$ such that $\\set{f^n|_U : n \\ge 0}$ is a [[normal family|JTVWRXXAKM]].\nNormality refers to maps into $\\hat{\\C}$ with the spherical metric.\nSo every sequence in the family has a subsequence that converges locally uniformly on $U$, possibly to the constant $\\infty$.\nA connected component of $F(f)$ is a *Fatou component*.\n\n## Remarks\n\nBy definition $F(f)$ is open.\nExample: for $f(z) = z^2$, the open unit disc lies in $F(f)$, since there $f^n(z) = z^{2^n} \\to 0$ locally uniformly.\nNon-example: $z = 1$ is not in $F(z^2)$; every neighbourhood of $1$ contains points tending to $0$ and points tending to $\\infty$, so no subsequence of iterates has a continuous limit there.\nThe Fatou set is completely invariant: $z \\in F(f)$ if and only if $f(z) \\in F(f)$.\nThe same definition with $\\C$ in place of $\\hat{\\C}$ is used for transcendental entire functions.\n",
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    "name": "Fatou set",
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    "tag": "REER3CG7DS",
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  "markdown": "https://tamnd.github.io/math2/node/REER3CG7DS.md",
  "msc": [
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  "name": "Fatou set",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/REE/REER3CG7DS.md",
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  "tag": "REER3CG7DS",
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