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 $2 πθ$ 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 $[a, b] \subset [0, 1]$ . 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]
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^[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 $2 πα$ 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 $2 πθ$ until the first time the orbit lands in $J$ ; then switch to the corresponding point in the second circle, rotate by $2 πθ$ 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. Bibcode:1968PNAS...60.1163V. doi:10.1073/pnas.60.4.1163. PMC 224897. PMID 16591677.
^ Masur, Howard; Tabachnikov, Serge (2002). "Rational Billiards and Flat Structures". In Hasselblatt, B.; Katok, A. (eds.). Handbook of Dynamical Systems (PDF). Vol. IA. Elsevier.