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

\phi:([0,1],+) \to (S^1, \cdot) \quad \phi(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 $x \isin [0,1]$ . 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 $θ$ 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 element of $[0,1]$ under the rotation $T θ$ is dense in $[0,1]$ . Therefore, irrational rotations are topologically transitive.
If $θ$ is irrational, then $T θ$ is uniquely ergodic.
Irrational (and rational) rotations are not topologically mixing.
Irrational rotations are ergodic with respect to the Lebesgue measure.
Irrational rotations are uniquely ergodic, with the Lebesgue measure serving as the unique invariant probability measure.
Suppose $[a, b] \subset [0,1]$ . 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 $\pi (t)=t \bmod 1$ .^[1]

Applications

Skew Products over Rotations of the Circle: In 1969^[2] 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 $2 π α$ 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 $2 π θ$ until the first time the orbit lands in $J$ ; then switch to the corresponding point in the second circle, rotate by $2 π θ$ 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."^[3]

References

↑ Fisher, Todd (2007). "Circle Homomorphisms" (PDF).
↑ Veech, William (August 1968). "A Kronecker-Weyl Theorem Modulo 2". Proceedings of the National Academy of Sciences. 60 (4): 1163–1164. PMC 224897 . PMID 16591677. doi:10.1073/pnas.60.4.1163.
↑ Masur, Howard; Tabachnikov, Serge (2002). "Rational Billiards and Flat Structures". In Hasselblatt, B.; Katok, A. Handbook of Dynamical Systems (PDF). IA. Elsevier.