Arnold tongue

Remarks

Example: the tongue $T_0$. The 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 $x$. So $T_0 = \set{(\Omega, K) : \abs{\Omega} \le K / (2\pi)}$, a wedge with its tip at $(0, 0)$. The point $(0.3, 0.5)$ is not in $T_0$, since $0.3 > 0.5/(2\pi) \approx 0.08$.

For $K = 0$ the map is the rotation by $\Omega$, so $T_{p/q}$ meets the line $K = 0$ only at $\Omega = p/q$. For $0 < K \le 1$ the section of $T_{p/q}$ at height $K$ is a closed interval of positive length. For fixed $K$, the function $\Omega \mapsto \rho(F_{\Omega, K})$ is continuous and nondecreasing; for $0 < K \le 1$ it is constant on each such interval and is a devil's staircase, like the Cantor function. Translation by $1$ in $\Omega$ maps $T_{p/q}$ onto $T_{p/q + 1}$.