Irrational rotation
# Irrational rotation

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

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

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

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

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

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

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 $\theta ={\frac {a}{b}}$ and $\gcd(a,b)=1$, then $T_{\theta }^{b}(x)=x$ when $x\in [0,1]$. It can also be shown that $T_{\theta }^{i}(x)\neq x$ when $1\leq i<b$.

## Significance

Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving $C^2-diffeomorphism$ of the circle with an irrational rotation number θ is topologically conjugate to $T_θ.$ An irrational rotation is a measure-preserving ergodic transformation, but it is not mixing. The Poincaré map for the dynamical system associated with the Kronecker foliation on a torus with angle θ> is the irrational rotation by θ. C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

## Properties

* If θ is irrational, then the orbit of any element of [0, 1] under the rotation $T_θ$ is dense in [0, 1]. Therefore, irrational rotations are topologically transitive.
* Irrational (and rational) rotations are not topologically mixing.
* Irrational rotations are uniquely ergodic, with the Lebesgue measure serving as the unique invariant probability measure.
* Suppose [a, b] ⊂ [0, 1]. Since $T_θ$ is ergodic, ${\text{lim}}_{N\to \infty }{\frac {1}{N}}\sum _{n=0}^{N-1}\chi _{[a,b)}(T_{\theta }^{n}(t))=b-a$.

## Generalizations

* Circle rotations are examples of group translations.
* For a general orientation preserving homomorphism f of $S^1$ to itself we call a homeomorphism $F:\mathbb {R} \to \mathbb {R}$ a lift of f if $\pi \circ F=f\circ \pi$ where $\pi (t)=t{\bmod {1}}$.
* The circle rotation can be thought of as a subdivision of a circle into two parts, which are then exchanged with each other. A subdivision into more than two parts, which are then permuted with one-another, is called an interval exchange transformation.
* Rigid rotations of compact groups effectively behave like circle rotations; the invariant measure is the Haar measure.

## Applications

* Skew Products over Rotations of the Circle: In 1969 William A. Veech constructed examples of minimal and not uniquely ergodic dynamical systems as follows: "Take two copies of the unit circle and mark off segment J of length 2πα in the counterclockwise direction on each one with endpoint at 0. Now take θ irrational and consider the following dynamical system. Start with a point p, say in the first circle. Rotate counterclockwise by 2πθ until the first time the orbit lands in J; then switch to the corresponding point in the second circle, rotate by 2πθ until the first time the point lands in J; switch back to the first circle and so forth. Veech showed that if θ is irrational, then there exists irrational α for which this system is minimal and the Lebesgue measure is not uniquely ergodic."
