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In mathematical theory of [[dynamical system]]s, an '''irrational rotation''' is a [[function (mathematics)|map]]
 
: <math>T_\theta : [0,1] \rightarrow [0,1],\quad T_\theta(x) \triangleq x + \theta \mod 1, </math>
 
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 ''&theta;'' of a full revolution (i.e., an angle of 2''&pi;θ'' radians). Since ''&theta;'' is irrational, the rotation has infinite [[Order (group theory)|order]] in the [[circle group]] and the map ''T''<sub>''&theta;''</sub> has no [[periodic orbit]]s. Moreover, the orbit of any point ''x'' under the [[iterated function|iterates]] of ''T''<sub>''&theta;''</sub>,
 
:<math>\{x+n \theta : n \in \mathbb{Z}\},</math>
 
is [[dense set|dense]] in the interval [0,&nbsp;1) or the circle.  
== Significance ==
 
Irrational rotations form a fundamental example in the theory of [[dynamical system]]s. According to the [[Denjoy theorem]], every orientation-preserving ''C''<sup>2</sup>-diffeomorphism of the circle with an irrational [[rotation number]] ''&theta;'' is [[topologically conjugate]] to ''T''<sub>''&theta;''</sub>.
An irrational rotation is a [[measure-preserving transformation|measure-preserving]] [[ergodic transformation]], but it is not [[mixing (physics)|mixing]]. The [[Poincaré map]] for the dynamical system associated with the [[Foliation#Examples|Kronecker foliation]] on a [[torus]] with angle ''θ'' is the irrational rotation by ''θ''. [[C*-algebra]]s associated with irrational rotations, known as [[irrational rotation algebra]]s, have been extensively studied.
 
==See also==
 
*[[Bernoulli map]]
*[[Modular arithmetic]]
*[[Siegel disc]]
*[[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
 
[[Category:Dynamical systems]]

Revision as of 17:31, 8 January 2014

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

Tθ:[0,1][0,1],Tθ(x)x+θmod1,

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+nθ:n},

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