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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 $2$.\nA *Siegel disc* of $f$ is a component $U$ of the [[Fatou set|C0N0XGC6GK]] of $f$ with the following properties.\n\n1. There is $p \\ge 1$ with $f^p(U) = U$.\n2. There is a [[conformal map|0YYT36R4YD]] $\\varphi \\colon U \\to \\mathbb{D}$ onto the [[unit disc|8G5ABXRK85]] $\\mathbb{D}$.\n3. There is an [[irrational number|0DZXAXEJQ9]] $\\theta$ with $\\varphi(f^p(z)) = e^{2\\pi i \\theta} \\varphi(z)$ for all $z \\in U$.\n\n## Remarks\n\nThe point $z_0 = \\varphi^{-1}(0)$ is a [[periodic point|T75CZ5H47N]] of $f$, and its multiplier is $(f^p)'(z_0) = e^{2\\pi i \\theta}$.\nSo the periodic point is neutral and irrationally indifferent, and the map is linearizable near it.\nExample: Siegel proved in 1942 that $f(z) = e^{2\\pi i \\theta} z + z^2$ is linearizable at $0$ when $\\theta$ is Diophantine, for instance the golden mean $(\\sqrt{5} - 1)/2$.\nThe component of the Fatou set containing $0$ is then a Siegel disc.\nNon-example: the unit disc for $f(z) = z^2$ fails clause 3, since $f$ is not injective there, while a rotation is.\nThe analogous domain with an annulus in place of the disc is a Herman ring.\n",
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