Irrational rotation

In the mathematical theory of dynamical systems, an irrational rotation is a map

${\displaystyle T_{\theta }:[0,1]\rightarrow [0,1],\quad T_{\theta }(x)\triangleq x+\theta \mod 1,}$

where θ is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion θ of a full revolution (i.e., an angle of 2πθ radians). Since θ is irrational, the rotation has infinite order in the circle group and the map T_θ has no periodic orbits.

Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map

${\displaystyle T_{\theta }:S^{1}\to S^{1},\quad \quad \quad T_{\theta }(x)=xe^{2\pi i\theta }}$

The relationship between the additive and multiplicative notations is the group isomorphism

${\displaystyle \varphi :([0,1],+)\to (S^{1},\cdot )\quad \varphi (x)=xe^{2\pi i\theta }}$.

It can be shown that φ is an isometry.

There is a strong distinction in circle rotations that depends on whether θ is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if ${\displaystyle \theta ={\frac {a}{b}}}$ and ${\displaystyle \gcd(a,b)=1}$, then ${\displaystyle T_{\theta }^{b}(x)=x}$ when ${\displaystyle x\in [0,1]}$. It can also be shown that ${\displaystyle T_{\theta }^{i}(x)\neq x}$ when ${\displaystyle 1\leq i<b}$.