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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 with the boundary points glued together, this map becomes a rotation of a circle by a proportion θ of a full revolution. Since θ is irrational, the rotation has infinite order in the circle group and the map Tθ 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
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