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  "body": "\n## Statement\n\nLet $f \\colon \\hat{\\C} \\to \\hat{\\C}$ be a [[rational function|0XDK9CGYKJ]] of degree at least $2$.\nLet $U$ be a component of the [[Fatou set|C0N0XGC6GK]] of $f$ with $f^p(U) = U$ for some $p \\ge 1$, and let $g = f^p$.\nThen exactly one of the following holds.\n\n1. Attracting case: $U$ contains a fixed point $z_0$ of $g$ with $\\abs{g'(z_0)} \u003c 1$, and $g^n \\to z_0$ locally uniformly on $U$.\n2. Parabolic case: $\\partial U$ contains a fixed point $z_0$ of $g$ with $g'(z_0) = 1$, and $g^n \\to z_0$ locally uniformly on $U$.\n3. Siegel case: $U$ is a [[Siegel disc|KNDYA9T8ZK]].\n4. Herman case: $U$ is a [[Herman ring|HZX6Q514X8]].\n\n## Remarks\n\nIn the first two cases $z_0$ is a [[periodic point|T75CZ5H47N]] of $f$, and the superattracting case $g'(z_0) = 0$ is included in the first case.\nIn the last two cases $g$ restricted to $U$ is conformally conjugate to an irrational rotation of a disc or of a round annulus.\nThe proof uses the theory of normal families: limits of subsequences of $g^n$ on $U$ are either constant, giving the first two cases, or not, giving the last two.\nFor $f(z) = z^2$, the unit disc is an attracting component with $z_0 = 0$.\nFor $f(z) = z + z^2$, the fixed point $0$ has multiplier $1$ and lies on the boundary of a parabolic component.\nBy [[Sullivan's theorem|C9FDR4BGMG]] every Fatou component is eventually periodic, so these four types describe the long-run behaviour on the whole Fatou set.\n\n## Explanations\n\n### L1 human\n\nA periodic piece of the Fatou set can behave in only four ways.\nIts points can be sucked into a fixed point inside it, or creep towards a fixed point on its edge.\nOr the piece can be a disc or a ring on which the map simply turns, like a wheel by an irrational angle.\nNothing else can happen.\n\n### L2 human\n\nThe theorem is the starting point for describing the Fatou set of any rational map.\nAttracting and parabolic cycles each attract at least one critical point.\nSo a map of degree $d$ has at most $2d - 2$ such cycles, the number of its critical points counted with multiplicity.\nSiegel discs and Herman rings are rotation domains; their boundaries lie in the closure of the critical orbits.\nPolynomials have no Herman rings, so for them only three cases occur.\n",
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