Irrational rotation

In mathematics, an irrational rotation is a map:$r\; :\; [0,1]\; ightarrow\; [0,1]$ given by

: $r(x)\; =\; x\; +\; heta\; mod\; 1$

(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:$\{x+n\; heta\; :\; n\; in\; mathbb\{Z\}\}$is dense in the circle. The orbit indeed cannot be periodic because if its period is "p" then "p"θ=0 mod 1 that means "p"θ="k" (integer) and θ would be rational. So the orbit must be infinite. If we consider a subdivision of the unit interval into "N" subintervals whose length is 1/"N", by the pigeon hole principle there must be at least a subinterval containing at least 2 points "x"+"a"θ and "x"+"b"θ of the orbit. This means that ("a"-"b")θ is smaller than 1/"N": the iteration of the map for a certain number of times provides a rotation smaller than 1/"N". Since "N" can be fixed to be arbitrarily large density follows.

Irrational rotations have much use in C* algebras and dynamical systems.

ee also

*Bernoulli map
*Circle map
*Denjoy diffeomorphism
*Ergodic system
*Irrational rotation algebra
*Toeplitz algebra

External links

* [http://www.math.harvard.edu/archive/118r_spring_05/handouts/symbolic.pdf One hard-to-read reference]
* [http://secamlocal.ex.ac.uk/people/staff/mph204/research_topics.html One reference; need a better one]