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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 $2$ on the [[Riemann sphere|0J78KBPFZB]].\nLet $U$ be a connected component of the [[Fatou set|C0N0XGC6GK]] of $f$.\nThen $U$ is eventually periodic: there are integers $m \\ge 0$ and $p \\ge 1$ with $f^{m+p}(U) = f^m(U)$.\n\n## Remarks\n\nA Fatou component that is not eventually periodic is called a wandering domain, so the theorem says that rational maps have none.\nThe proof, due to Dennis Sullivan, uses [[quasiconformal mappings|SB8KG46NHP]].\nA wandering domain would support an infinite-dimensional family of quasiconformal deformations of $f$.\nEach deformation is again a rational map of the same degree, and these maps form a finite-dimensional space, a contradiction.\nThe hypothesis that $f$ is rational is needed.\nThe transcendental [[entire function|0G4F0XB9X2]] $g(z) = z + \\sin z + 2\\pi$ has wandering domains.\nIndeed $h(z) = z + \\sin z$ has superattracting fixed points at the odd multiples of $\\pi$, and $g = h + 2\\pi$ moves each of their basins to the next one.\n\n## Explanations\n\n### L1 human\n\nThe Fatou set of a rational map breaks into pieces, and the map moves these pieces around.\nThe theorem says that no piece can wander forever through new pieces.\nAfter finitely many steps every piece falls into a cycle of pieces that repeats.\nThis reduces the study of the Fatou set to its periodic pieces, whose types are known.\n\n### L2 human\n\nTogether with the classification of periodic Fatou components, the theorem gives a complete picture of the Fatou set of a rational map.\nEach component eventually lands on a cycle of attracting basins, parabolic basins, Siegel discs or Herman rings.\nThe method of proof, quasiconformal deformation, became a basic tool of the subject.\nFor transcendental entire functions wandering domains exist, and they remain an active topic of research.\n",
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  "name": "Sullivan's no wandering domains theorem",
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