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

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

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

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

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

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

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 ${\displaystyle \theta ={\frac {a}{b))}$ and ${\displaystyle \gcd(a,b)=1}$, then ${\displaystyle T_{\theta }^{b}(x)=x}$ when ${\displaystyle x\in [0,1]}$. It can also be shown that ${\displaystyle T_{\theta }^{i}(x)\neq x}$ when ${\displaystyle 1\leq i<b}$.

Significance

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

Generalizations

• Circle rotations are examples of group translations.
• For a general orientation preserving homomorphism f of S¹ to itself we call a homeomorphism ${\displaystyle F:\mathbb {R} \to \mathbb {R} }$ a lift of f if ${\displaystyle \pi \circ F=f\circ \pi }$ where ${\displaystyle \pi (t)=t{\bmod {1))}$.^[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 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."^[3]