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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 *Herman ring* 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 are $r \u003e 1$ and a [[conformal map|0YYT36R4YD]] $\\varphi$ from $U$ onto the round [[annulus|0CW52NV5TC]] $A_r = \\set{w \\in \\C : 1 \u003c \\abs{w} \u003c r}$.\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 circles $\\abs{w} = s$ with $1 \u003c s \u003c r$ pull back to invariant closed curves of $f^p$ in $U$.\nNon-example: a polynomial $P$ has no Herman ring.\nIf $U$ were one, an invariant curve $\\gamma$ in $U$ would bound a disc $D$, and by the maximum principle the iterates of $P$ would be bounded on $D$.\nThen $D$ would lie in the Fatou set, so $U$ could not be a component of it with a hole.\nHerman constructed the first examples of rational maps with Herman rings.\nShishikura later built them by quasiconformal surgery from maps with Siegel discs.\n",
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