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 $&theta;$ 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 $&theta;$ of a full revolution (i.e., an angle of $2&pi;&theta;$ radians). Since $&theta;$ is irrational, the rotation has infinite order in the circle group and the map $T_{&theta;}$ 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 $&phi;$ is an isometry.

There is a strong distinction in circle rotations that depends on whether $&theta;$ 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 $$x \isin [0,1]$$. It can also be shown that $$T_\theta^i(x) \ne x$$ when $$1 \le i < b$$.

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 $&theta;$ is topologically conjugate to $T_{&theta;}$. 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 $&theta;>$ is the irrational rotation by $&theta;$. C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

Properties

 * If $&theta;$ is irrational, then the orbit of any element of $[0, 1]$ under the rotation $T_{&theta;}$ 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 $[a, b] &sub; [0, 1]$. Since $T_{&theta;}$ 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 $2&pi;&alpha;$ in the counterclockwise direction on each one with endpoint at 0. Now take $&theta;$ irrational and consider the following dynamical system. Start with a point $p$, say in the first circle. Rotate counterclockwise by $2&pi;&theta;$ until the first time the orbit lands in $J$; then switch to the corresponding point in the second circle, rotate by $2&pi;&theta;$ until the first time the point lands in $J$; switch back to the first circle and so forth. Veech showed that if $&theta;$ is irrational, then there exists irrational $&alpha;$ for which this system is minimal and the Lebesgue measure is not uniquely ergodic."