Irrational Rotation

In 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 (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. Moreover, the orbit of any point x under the iterates of T_θ,

is dense in the interval [0, 1) or the circle.