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| In mathematical theory of [[dynamical system]]s, an '''irrational rotation''' is a [[function (mathematics)|map]]
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| : <math>T_\theta : [0,1] \rightarrow [0,1],\quad T_\theta(x) \triangleq x + \theta \mod 1, </math>
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| 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 2''πθ'' radians). Since ''θ'' is irrational, the rotation has infinite [[Order (group theory)|order]] in the [[circle group]] and the map ''T''<sub>''θ''</sub> has no [[periodic orbit]]s. Moreover, the orbit of any point ''x'' under the [[iterated function|iterates]] of ''T''<sub>''θ''</sub>,
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| :<math>\{x+n \theta : n \in \mathbb{Z}\},</math> | |
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| is [[dense set|dense]] in the interval [0, 1) or the circle.
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| == Significance ==
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| 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]] ''θ'' is [[topologically conjugate]] to ''T''<sub>''θ''</sub>.
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| 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.
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| ==See also==
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| *[[Bernoulli map]]
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| *[[Modular arithmetic]]
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| *[[Siegel disc]]
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| *[[Toeplitz algebra]]
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| *[[Phase locking]] (circle map)
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| == References ==
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| * C. E. Silva, ''Invitation to ergodic theory'', Student Mathematical Library, vol 42, [[American Mathematical Society]], 2008 ISBN 978-0-8218-4420-5
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| [[Category:Dynamical systems]]
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Hello, my name is Andrew and my wife doesn't like it at all. He works as a bookkeeper. As a woman what she really likes is style and she's been doing it for fairly a whilst. For years he's been living in Alaska and he doesn't strategy on altering it.
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