|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In mathematical theory of [[dynamical system]]s, an '''irrational rotation''' is a [[function (mathematics)|map]]
| | Wilber Berryhill is what his wife loves to call him and he completely loves this title. She is truly fond of caving but she doesn't have the time lately. Distributing production has been his occupation for some time. Some time ago love psychics ([http://breenq.com/index.php?do=/profile-1144/info/ http://breenq.com/index.php?do=/profile-1144/info/]) she selected to reside in Alaska and her mothers and fathers reside close by.<br><br>my webpage ... phone [http://www.chk.woobi.co.kr/xe/?document_srl=346069 real psychic] readings ([http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ you can try these out]) |
| | |
| : <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 ''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>,
| |
| | |
| :<math>\{x+n \theta : n \in \mathbb{Z}\},</math> | |
| | |
| is [[dense set|dense]] in the interval [0, 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]] ''θ'' is [[topologically conjugate]] to ''T''<sub>''θ''</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]]
| |
Wilber Berryhill is what his wife loves to call him and he completely loves this title. She is truly fond of caving but she doesn't have the time lately. Distributing production has been his occupation for some time. Some time ago love psychics (http://breenq.com/index.php?do=/profile-1144/info/) she selected to reside in Alaska and her mothers and fathers reside close by.
my webpage ... phone real psychic readings (you can try these out)