Irrational rotation
In mathematical theory of dynamical systems, an irrational rotation is a map
![T_\theta�: [0,1] \rightarrow [0,1],\quad r(x) = x %2B \theta \mod 1,](/2012-wikipedia_en_all_nopic_01_2012/I/88c3aa532716b8e024d0dfd8ca939eb8.png)
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 2πθ radians). Since θ is irrational, the rotation has infinite order in the circle group and the map Tθ has no periodic orbits. Moreover, the orbit of any point x under the iterates of Tθ,
is dense in the interval [0, 1) or the circle.
Significance
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. 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.
See also
- Bernoulli map
- Modular arithmetic
- Toeplitz algebra
- Phase locking (Circle Map)
References
- C.E.Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008 ISBN 978-0-8218-4420-5