Irrational rotation

Short description: Rotation of a circle by an angle of π times an irrational number

Sturmian sequence generated by irrational rotation with theta=0.2882748715208621 and x=0.078943143

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

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

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

[math]\displaystyle{ T_\theta :S^1 \to S^1, \quad \quad \quad T_\theta(x)=xe^{2\pi i\theta} }[/math]

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

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

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 [math]\displaystyle{ \theta = \frac{a}{b} }[/math] and [math]\displaystyle{ \gcd(a,b) = 1 }[/math], then [math]\displaystyle{ T_\theta^b(x) = x }[/math] when [math]\displaystyle{ x \isin [0,1] }[/math]. It can also be shown that [math]\displaystyle{ T_\theta^i(x) \ne x }[/math] when [math]\displaystyle{ 1 \le i \lt b }[/math].

Significance

Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving C²-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,
  [math]\displaystyle{ \text{lim} _ {N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} \chi_{[a,b)}(T_\theta ^n (t))=b-a }[/math].

Generalizations

• Circle rotations are examples of group translations.
• For a general orientation preserving homomorphism f of S¹ to itself we call a homeomorphism [math]\displaystyle{ F:\mathbb{R}\to \mathbb{R} }[/math] a lift of f if [math]\displaystyle{ \pi \circ F=f \circ \pi }[/math] where [math]\displaystyle{ \pi (t)=t \bmod 1 }[/math].
• 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."

See also

• Bernoulli map
• Modular arithmetic
• Siegel disc
• Toeplitz algebra
• Phase locking (circle map)

Further reading

• C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008 ISBN:978-0-8218-4420-5