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 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 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] ⊂ [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 S¹ to itself we call a homeomorphism $F:\mathbb{R}\to \mathbb{R}$ a lift of f if $\pi \circ F=f \circ \pi$ where $\pi (t)=t \bmod 1$.

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