---
tag: 1G2ZRM270V
kind: definition
name: Arnold tongue
msc: ["37E45", "37E10", "37G15"]
defines: WQF5ASXZTH
uses: [N2SVK69S53, DP4WWQBJJF, NPBYYK63FE]
sources:
  - url: https://en.wikipedia.org/wiki/Arnold_tongue
---

## Definition

For $\Omega \in \R$ 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.
Let $F_{\Omega, K}(x) = x + \Omega - \frac{K}{2\pi} \sin(2\pi x)$ 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]].
For a rational number $p/q$, the *Arnold tongue* of $p/q$ is the set
$$T_{p/q} = \set{(\Omega, K) \in \R \times [0, 1] : \rho(F_{\Omega, K}) = p/q}.$$

## 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|NPBYYK63FE]].
Translation by $1$ in $\Omega$ maps $T_{p/q}$ onto $T_{p/q + 1}$.

## Explanations

### L1 human

Each rational number $p/q$ has its tongue: the set of settings of the circle map where the average turn per step is exactly $p/q$.
The example $T_0$ shows the typical shape: a wedge whose width grows with the strength $K$ of the kick.