Irrational rotation

In mathematics, an irrational rotation is a map

given by

(see modular arithmetics) where θ is an irrational number. The name comes from the fact that this map comes from a rotation by an angle of θ on a circle after identifying that circle with the interval [0, 1] where the boundary points are identified (that is R/Z).

Such a rotation is an element of infinite order in the circle group. If θ were rational, then the rotation would be an element of finite order. In other words, if θ were rational, then applying the rotation a sufficient number of times would map all elements of the circle back on to themselves.

Given any starting point this will generate a dense set in the interval [0, 1) by repeatedly applying the mapping r to it as an iterated function. In other words for any x the set

is dense in the circle. The orbit indeed cannot be periodic because if its period is p then pθ=0 mod 1 that means pθ=k (integer) and θ would be rational. So the orbit must be infinite.

Irrational rotations have much use in C* algebras and dynamical systems. It can be shown that irrational rotation is ergodic, but 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.

See also

• Bernoulli map
• Denjoy diffeomorphism
• Ergodic system
• Irrational rotation algebra
• Rotation number
• Toeplitz algebra

References

• C.E.Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008 ISBN 978-0-8218-4420-5

External links

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