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  "body": "\n## Definition\n\nLet $S^1 = \\R / \\Z$ be the [[circle|060W5CBZP6]], with projection $\\pi \\colon \\R \\to S^1$, and let $f \\colon S^1 \\to S^1$ be an orientation-preserving [[homeomorphism|0QT00PB5G2]].\nA *lift* of $f$ is a continuous map $F \\colon \\R \\to \\R$ with $\\pi \\circ F = f \\circ \\pi$.\nEvery such lift is increasing and satisfies $F(x + 1) = F(x) + 1$, and two lifts differ by an integer constant.\nThe *translation number* of a lift $F$ is the limit\n$$\\rho(F) = \\lim_{n \\to \\infty} \\frac{F^n(x) - x}{n},$$\nwhere $F^n$ is the $n$-th iterate of $F$.\nThe limit exists and does not depend on $x \\in \\R$.\nThe *rotation number* of $f$ is $\\rho(f) = \\rho(F) - \\floor{\\rho(F)} \\in [0, 1)$, which does not depend on the choice of lift $F$.\n\n## Remarks\n\nExample: the rotation $R_\\alpha(x) = x + \\alpha$ of $S^1$ has the lift $F(x) = x + \\alpha$, and $F^n(x) - x = n\\alpha$.\nSo $\\rho(F) = \\alpha$, and $\\rho(R_\\alpha)$ is the fractional part of $\\alpha$.\n\nNon-example: the doubling map $x \\mapsto 2x$ of $S^1$ is not a homeomorphism.\nIts lift $F(x) = 2x$ satisfies $F(x + 1) = F(x) + 2$, and $(F^n(x) - x)/n = (2^n - 1) x / n$ has no finite limit for $x \\neq 0$, so no rotation number is defined.\n\nThe rotation number is invariant under conjugacy: if $h$ is an orientation-preserving homeomorphism of $S^1$, then $\\rho(h^{-1} \\circ f \\circ h) = \\rho(f)$.\nIt depends continuously on $f$ in the uniform topology.\nThe fractional part is often left out, and $\\rho(f)$ is viewed as an element of $\\R / \\Z$.\n",
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