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  "body": "\n## Definition\n\nLet $\\Omega \\in \\R$ and $K \\ge 0$.\nLet $F_{\\Omega, K} \\colon \\R \\to \\R$ be the map\n$$F_{\\Omega, K}(x) = x + \\Omega - \\frac{K}{2\\pi} \\sin(2\\pi x).$$\nSince $F_{\\Omega, K}(x + 1) = F_{\\Omega, K}(x) + 1$, it induces a map $f_{\\Omega, K}$ of the [[circle|060W5CBZP6]] $\\R / \\Z$, given by $f_{\\Omega, K}(x + \\Z) = F_{\\Omega, K}(x) + \\Z$.\nThe map $f_{\\Omega, K}$ is the *(standard) circle map* with parameters $\\Omega$ and $K$, and $F_{\\Omega, K}$ is its lift.\n\n## Remarks\n\nExample: for $K = 0$ the circle map is the rotation $x \\mapsto x + \\Omega$; for irrational $\\Omega$ it is an [[irrational rotation|2Q1J7RZ4TK]].\n\nThe derivative is $F'_{\\Omega, K}(x) = 1 - K \\cos(2\\pi x)$.\nFor $0 \\le K \u003c 1$ it is positive everywhere, so $f_{\\Omega, K}$ is a [[diffeomorphism|0KZZ70ZHXP]] of the circle.\nFor $K = 1$ the derivative vanishes at $x = 0$, so $f_{\\Omega, 1}$ is a [[homeomorphism|0QT00PB5G2]] with a [[critical point|0YTCMFB4HC]], but not a diffeomorphism.\n\nNon-example: for $K \u003e 1$ the derivative takes negative values, so $F_{\\Omega, K}$ is not monotone and $f_{\\Omega, K}$ is not invertible.\nThe map is still defined, but it is not a homeomorphism.\n\nThe parameter $\\Omega$ is often taken in $[0, 1)$, since $f_{\\Omega + 1, K} = f_{\\Omega, K}$.\n",
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