In the mathematical theory of dynamical systems, an irrational rotation is a map where is an irrational number. Under the identification of a circle with , or with the interval with the boundary points glued together, this map becomes a rotation of a circle by a proportion of a full revolution (i.e., an angle of radians). Since is irrational, the rotation has infinite order in the circle group and the map 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 is an isometry. There is a strong distinction in circle rotations that depends on whether 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 . Significance Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving -diffeomorphism of the circle with an irrational rotation number is topologically conjugate to . 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 is the irrational rotation by . C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied. Properties If is irrational, then the orbit of any element of under the rotation is dense in . 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 . Since is ergodic,. Generalizations Circle rotations are examples of group translations. For a general orientation preserving homomorphism of to itself we call a homeomorphism a lift of if where . 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