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

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 2*πθ* 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

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

- .

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 and , then when . It can also be shown that
when .

Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving *C*^{2}-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.

- 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,

.

- Circle rotations are examples of group translations.
- For a general orientation preserving homomorphism
*f*of*S*^{1}to itself we call a homeomorphism a*lift*of*f*if where .^{[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.

- 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 2*πα*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 2*πθ*until the first time the orbit lands in*J*; then switch to the corresponding point in the second circle, rotate by 2*πθ*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]}