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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]}

**^**Fisher, Todd (2007). "Circle Homomorphisms" (PDF).**^**Veech, William (August 1968). "A Kronecker-Weyl Theorem Modulo 2".*Proceedings of the National Academy of Sciences*.**60**(4): 1163–1164. Bibcode:1968PNAS...60.1163V. doi:10.1073/pnas.60.4.1163. PMC 224897. PMID 16591677.**^**Masur, Howard; Tabachnikov, Serge (2002). "Rational Billiards and Flat Structures". In Hasselblatt, B.; Katok, A. (eds.).*Handbook of Dynamical Systems*(PDF).**IA**. Elsevier.

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