Irrational rotation

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

An irrational rotation in the mathematical theory of dynamical systems, is defined as a map $f$, as follows:

Let $x\in [0,1)$ be arbitrary, and let $\theta$ be any irrational number. For $i\in$ Template:Math define $f_{\theta }^i:[0,1)\to [0,1)$ by

$f_{\theta }^i(x)=x+i\cdot \theta$ (mod 1)

Under the identification of a circle with Template:Math, or with the interval Template:Math with the boundary points glued together, this map becomes a rotation of a circle by a proportion Template:Math of a full revolution (i.e., an angle of Template:Math radians). Since Template:Math is irrational, the rotation has infinite order in the circle group, and the map Template:Math has no periodic orbits.

Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map

$f_{\theta }:S^1\to S^1,f_{\theta }(x)=x{e}^{2\pi i\theta }$

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

$\phi :([0,1),+)\to (S^1,\cdot )\phi (x)=x{e}^{2\pi i\theta }$.

It can be shown that Template:Math is an isometry.

There is a strong distinction in circle rotations that depends on whether Template:Math 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 $f_{\theta }^b(x)=x$ when $x\in [0,1)$. It can also be shown that $f_{\theta }^i(x)\ne x$ 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 Template:Math-diffeomorphism of the circle with an irrational rotation number Template:Math is topologically conjugate to Template:Math. 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 Template:Math is the irrational rotation by Template:Math. C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

Properties

• If Template:Math is irrational, then the orbit of any element of Template:Math under the rotation Template:Math is dense in Template:Math. 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 Template:Math. Since Template:Math is ergodic,
  ${\text{lim}}_{N\to \mathrm{\infty }}\frac{1}{N}\sum _{n=0}^{N-1}{\chi }_{[a,b)}(f_{\theta }^n(t))=b-a$.

Generalizations

• Circle rotations are examples of group translations.
• For a general orientation preserving homomorphism Template:Math of Template:Math to itself we call a homeomorphism $F:ℝ\to ℝ$ a lift of Template:Math if $\pi \circ F=f\circ \pi$ where $\pi (t)=tmod1$.
• 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 Template:Math of length Template:Math in the counterclockwise direction on each one with endpoint at 0. Now take Template:Math irrational and consider the following dynamical system. Start with a point Template:Math, say in the first circle. Rotate counterclockwise by Template:Math until the first time the orbit lands in Template:Math; then switch to the corresponding point in the second circle, rotate by Template:Math until the first time the point lands in Template:Math; switch back to the first circle and so forth. Veech showed that if Template:Math is irrational, then there exists irrational Template:Math 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)
• Weyl sequence

Further reading

• C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008 Template:ISBN