irrational rotation

{{Short description|Rotation of a circle by an angle of π times an irrational number}}

File:Sturmian-sequence-from-irrational-rotation.gif

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 {{math|θ}} is an irrational number. Under the identification of a circle with {{math|R/Z}}, or with the interval {{math|[0, 1]}} with the boundary points glued together, this map becomes a rotation of a circle by a proportion {{math|θ}} of a full revolution (i.e., an angle of {{math|2πθ}} radians). Since {{math|θ}} is irrational, the rotation has infinite order in the circle group and the map {{math|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 {{math|φ}} is an isometry.

There is a strong distinction in circle rotations that depends on whether {{math|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 {{math|C²}}-diffeomorphism of the circle with an irrational rotation number {{math|θ}} is topologically conjugate to {{math|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 {{math|θ>}} is the irrational rotation by {{math|θ}}. C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

Properties

If {{math|θ}} is irrational, then the orbit of any element of {{math|[0, 1]}} under the rotation {{math|T_θ}} is dense in {{math|[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 {{math|[a, b] ⊂ [0, 1]}}. Since {{math|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 {{math|f}} of {{math|S¹}} to itself we call a homeomorphism $F:\mathbb{R}\to \mathbb{R}$ a lift of {{math|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 {{math|J}} of length {{math|2πα}} in the counterclockwise direction on each one with endpoint at 0. Now take {{math|θ}} irrational and consider the following dynamical system. Start with a point {{math|p}}, say in the first circle. Rotate counterclockwise by {{math|2πθ}} until the first time the orbit lands in {{math|J}}; then switch to the corresponding point in the second circle, rotate by {{math|2πθ}} until the first time the point lands in {{math|J}}; switch back to the first circle and so forth. Veech showed that if {{math|θ}} is irrational, then there exists irrational {{math|α}} for which this system is minimal and the Lebesgue measure is not uniquely ergodic."

References

{{reflist|refs=

{{ cite web

|last=Fisher|first=Todd

|year=2007

|title=Circle Homomorphisms

|url=https://math.byu.edu/~tfisher/documents/classes/2008/winter/635/Lecture2.pdf

}}

{{cite journal

|last=Veech|first=William | authorlink = William A. Veech

|title=A Kronecker-Weyl Theorem Modulo 2

|journal=Proceedings of the National Academy of Sciences

|pmc=224897

|date=August 1968

|volume=60|number=4

|pages=1163–1164

|doi=10.1073/pnas.60.4.1163

|pmid=16591677

|bibcode=1968PNAS...60.1163V |doi-access=free }}

{{cite book

|last1=Masur|first1=Howard

|first2=Serge|last2=Tabachnikov|authorlink2=Sergei Tabachnikov

|chapter=Rational Billiards and Flat Structures

|url=http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf

|title=Handbook of Dynamical Systems

|volume=IA

|editor1-first=B.|editor1-last=Hasselblatt

|editor2-first=A.|editor2-last=Katok

|publisher=Elsevier

|year=2002

}}