Irrational rotation

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 2πθ radians).

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 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

Irrational rotations form a fundamental example in the theory of dynamical systems.

According to the Denjoy theorem, every orientation-preserving C2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to Tθ.