Irrational rotation

<h2 id="significance">Significance</h2>
Irrational rotations form a fundamental example in the theory of <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>. According to the <a href="/facts/Denjoy%2527s_theorem_on_rotation_number/2neCUcop">Denjoy theorem</a>, every orientation-preserving C2-diffeomorphism of the circle with an irrational <a href="/facts/Rotation_number/6sGrIcrB">rotation number</a> θ is <a href="/facts/Topologically_conjugate/vQvYmcIA">topologically conjugate</a> to Tθ. An irrational rotation is a <a href="/facts/Measure-preserving_transformation/acIdlpqt">measure-preserving</a> <a href="/facts/Ergodic_transformation/rkqCbIy0">ergodic transformation</a>, but it is not <a href="/facts/Mixing_(physics)/Mlt6XFoK">mixing</a>. The <a href="/facts/Poincar%25C3%25A9_map/pd15LII2">Poincaré map</a> for the dynamical system associated with the <a href="/facts/Foliation/cxml9vyJ">Kronecker foliation</a> on a <a href="/facts/Torus/J9SkaV1W">torus</a> with angle θ> is the irrational rotation by θ. <a href="/facts/C*-algebra/B3tnGHIo">C*-algebras</a> associated with irrational rotations, known as <a href="/facts/Noncommutative_torus/2ICPC2Iw">irrational rotation algebras</a>, have been extensively studied.

<h2 id="properties">Properties</h2>
<ul><li>If θ is irrational, then the orbit of any element of [0, 1] under the rotation Tθ is <a href="/facts/Dense_set/nPT9Yxao">dense</a> in [0, 1]. Therefore, irrational rotations are <a href="/facts/Topologically_transitive/ttsAJCID">topologically transitive</a>.</li>
<li>Irrational (and rational) rotations are not <a href="/facts/Topologically_mixing/ttsAJCID">topologically mixing</a>.</li>
<li>Irrational rotations are uniquely <a href="/facts/Ergodicity/CKY3Gr9v">ergodic</a>, with the Lebesgue measure serving as the unique invariant probability measure.</li>
<li>Suppose [a, b] ⊂ [0, 1]. Since Tθ is ergodic,
 
 
 
 
 
 lim
 
 
 N
 →
 ∞
 
 
 
 
 1
 N
 
 
 
 ∑
 
 n
 =
 0
 
 
 N
 −
 1
 
 
 
 χ
 
 [
 a
 ,
 b
 )
 
 
 (
 
 T
 
 θ
 
 
 n
 
 
 (
 t
 )
 )
 =
 b
 −
 a
 
 
 {\displaystyle {\text{lim}}_{N\to \infty }{\frac {1}{N}}\sum _{n=0}^{N-1}\chi _{[a,b)}(T_{\theta }^{n}(t))=b-a}
 
.</li></ul>
<h2 id="generalizations">Generalizations</h2>
<ul><li>Circle rotations are examples of <a href="/facts/Translation_(group_theory)/IQTYqINt">group translations</a>.</li>
<li>For a general orientation preserving homomorphism f of S1 to itself we call a homeomorphism 
 
 
 
 F
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle F:\mathbb {R} \to \mathbb {R} }
 
 a lift of f if 
 
 
 
 π
 ∘
 F
 =
 f
 ∘
 π
 
 
 {\displaystyle \pi \circ F=f\circ \pi }
 
 where 
 
 
 
 π
 (
 t
 )
 =
 t
 
 mod
 
 1
 
 
 
 
 {\displaystyle \pi (t)=t{\bmod {1}}}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>The circle rotation can be thought of as a subdivision of a circle into two parts, which are then exchanged with each other. A subdivision into more than two parts, which are then permuted with one-another, is called an <a href="/facts/Interval_exchange_transformation/G7CThewu">interval exchange transformation</a>.</li>
<li>Rigid rotations of <a href="/facts/Compact_group/Itl2HJaR">compact groups</a> effectively behave like circle rotations; the invariant measure is the <a href="/facts/Haar_measure/2IcmTcuU">Haar measure</a>.</li></ul>
<h2 id="applications">Applications</h2>
<ul><li>Skew Products over Rotations of the Circle: In 1969<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> <a href="/facts/William_A._Veech/ca4myBwR">William A. Veech</a> constructed examples of minimal and not uniquely ergodic dynamical systems as follows: "Take two copies of the unit circle and mark off segment J of length 2πα in the counterclockwise direction on each one with endpoint at 0. Now take θ irrational and consider the following dynamical system. Start with a point p, say in the first circle. Rotate counterclockwise by 2πθ until the first time the orbit lands in J; then switch to the corresponding point in the second circle, rotate by 2πθ until the first time the point lands in J; switch back to the first circle and so forth. Veech showed that if θ is irrational, then there exists irrational α for which this system is minimal and the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> is not uniquely ergodic."<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bernoulli_map/Hopsjd88">Bernoulli map</a></li>
<li><a href="/facts/Modular_arithmetic/0wtRa5dU">Modular arithmetic</a></li>
<li><a href="/facts/Siegel_disc/xTXJh5RV">Siegel disc</a></li>
<li><a href="/facts/Toeplitz_algebra/EBH9IINT">Toeplitz algebra</a></li>
<li><a href="/facts/Phase_locking/VxKBdhcW">Phase locking</a> (circle map)</li>
<li><a href="/facts/Weyl_sequence/uloBPqP0">Weyl sequence</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, 2008 <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4420-5</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Fisher, Todd (2007). "Circle Homomorphisms" (PDF). <a href="https://math.byu.edu/~tfisher/documents/classes/2008/winter/635/Lecture2.pdf" target="_blank">https://math.byu.edu/~tfisher/documents/classes/2008/winter/635/Lecture2.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Veech, William (August 1968). "A Kronecker-Weyl Theorem Modulo 2". Proceedings of the National Academy of Sciences. 60 (4): 1163–1164. Bibcode:1968PNAS...60.1163V. doi:10.1073/pnas.60.4.1163. PMC 224897. PMID 16591677. <a href="/wiki/William_A._Veech" target="_blank">/wiki/William_A._Veech</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Masur, Howard; Tabachnikov, Serge (2002). "Rational Billiards and Flat Structures". In Hasselblatt, B.; Katok, A. (eds.). Handbook of Dynamical Systems (PDF). Vol. IA. Elsevier. <a href="/wiki/Sergei_Tabachnikov" target="_blank">/wiki/Sergei_Tabachnikov</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Irrational rotation open-in-new

Irrational rotation