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  "body": "\n## Definition\n\nFor $\\Omega \\in \\R$ and $0 \\le K \\le 1$, let $f_{\\Omega, K}$ be the standard [[circle map|N2SVK69S53]] $x \\mapsto x + \\Omega - \\frac{K}{2\\pi} \\sin(2\\pi x)$ on $\\R / \\Z$, an orientation-preserving homeomorphism.\nLet $F_{\\Omega, K}(x) = x + \\Omega - \\frac{K}{2\\pi} \\sin(2\\pi x)$ be its lift to $\\R$, and let $\\rho(F_{\\Omega, K})$ be the translation number of this lift, as in the definition of the [[rotation number|DP4WWQBJJF]].\nFor a rational number $p/q$, the *Arnold tongue* of $p/q$ is the set\n$$T_{p/q} = \\set{(\\Omega, K) \\in \\R \\times [0, 1] : \\rho(F_{\\Omega, K}) = p/q}.$$\n\n## Remarks\n\nExample: the tongue $T_0$.\nThe translation number of $F_{\\Omega, K}$ is $0$ exactly when $F_{\\Omega, K}$ has a fixed point, that is, when $\\Omega = \\frac{K}{2\\pi} \\sin(2\\pi x)$ for some $x$.\nSo $T_0 = \\set{(\\Omega, K) : \\abs{\\Omega} \\le K / (2\\pi)}$, a wedge with its tip at $(0, 0)$.\nThe point $(0.3, 0.5)$ is not in $T_0$, since $0.3 \u003e 0.5/(2\\pi) \\approx 0.08$.\n\nFor $K = 0$ the map is the rotation by $\\Omega$, so $T_{p/q}$ meets the line $K = 0$ only at $\\Omega = p/q$.\nFor $0 \u003c K \\le 1$ the section of $T_{p/q}$ at height $K$ is a closed interval of positive length.\nFor fixed $K$, the function $\\Omega \\mapsto \\rho(F_{\\Omega, K})$ is continuous and nondecreasing; for $0 \u003c K \\le 1$ it is constant on each such interval and is a devil's staircase, like the [[Cantor function|NPBYYK63FE]].\nTranslation by $1$ in $\\Omega$ maps $T_{p/q}$ onto $T_{p/q + 1}$.\n\n## Explanations\n\n### L1 human\n\nEach rational number $p/q$ has its tongue: the set of settings of the circle map where the average turn per step is exactly $p/q$.\nThe example $T_0$ shows the typical shape: a wedge whose width grows with the strength $K$ of the kick.\n",
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