Irrational rotation

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In mathematical theory of dynamical systems, an irrational rotation is a map

T_{\theta }:[0,1]\rightarrow [0,1],\quad T_{\theta }(x)\triangleq x+\theta \mod 1,

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,

\{x+n\theta :n\in {\mathbb {Z}}\},

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

References

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