In the mathematical theory ofdynamical systems, anirrational rotation is amap

${\displaystyle T_{\theta }:[0,1]\to [0,1],T_{\theta }(x)\triangleq x+\theta \mathrm{mod}1,}$

whereθ is anirrational number. Under the identification of acircle withR/Z, or with the interval[0, 1] with the boundary points glued together, this map becomes arotation of acircle by a proportionθ of a full revolution (i.e., an angle of2πθ radians). Sinceθ is irrational, the rotation has infiniteorder in thecircle group and the mapT_θ has noperiodic orbits.

Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map

${\displaystyle T_{\theta }:S^1\to S^1,T_{\theta }(x)=x{e}^{2\pi i\theta }}$

The relationship between the additive and multiplicative notations is the group isomorphism

${\displaystyle \phi :([0,1],+)\to (S^1,\cdot )\phi (x)=x{e}^{2\pi i\theta }}$.

It can be shown thatφ is anisometry.

There is a strong distinction in circle rotations that depends on whetherθ is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if${\displaystyle \theta =\frac{a}{b}}$ and${\displaystyle gcd(a,b)=1}$, then${\displaystyle T_{\theta }^b(x)=x}$ when${\displaystyle x\in [0,1]}$. It can also be shown that${\displaystyle T_{\theta }^i(x)\ne x}$ when.

Irrational rotations form a fundamental example in the theory ofdynamical systems. According to theDenjoy theorem, every orientation-preservingC²-diffeomorphism of the circle with an irrationalrotation numberθ istopologically conjugate toT_θ. An irrational rotation is ameasure-preservingergodic transformation, but it is notmixing. ThePoincaré map for the dynamical system associated with theKronecker foliation on atorus with angleθ> is the irrational rotation byθ.C*-algebras associated with irrational rotations, known asirrational rotation algebras, have been extensively studied.

• Ifθ is irrational, then the orbit of any element of[0, 1] under the rotationT_θ isdense in[0, 1]. Therefore, irrational rotations aretopologically transitive.
• Irrational (and rational) rotations are nottopologically mixing.
• Irrational rotations are uniquelyergodic, with the Lebesgue measure serving as the unique invariant probability measure.
• Suppose[a,b] ⊂ [0, 1]. SinceT_θ is ergodic,
  ${\displaystyle {\text{lim}}_{N\to \mathrm{\infty }}\frac{1}{N}\sum _{n=0}^{N-1}{\chi }_{[a,b)}(T_{\theta }^n(t))=b-a}$.

• Circle rotations are examples ofgroup translations.
• For a general orientation preserving homomorphismf ofS¹ to itself we call a homeomorphism${\displaystyle F:\mathbb{R}\to \mathbb{R}}$ alift off if${\displaystyle \pi \circ F=f\circ \pi }$ where${\displaystyle \pi (t)=tmod1}$.^[1]
• 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 aninterval exchange transformation.
• Rigid rotations ofcompact groups effectively behave like circle rotations; the invariant measure is theHaar measure.

• Skew Products over Rotations of the Circle: In 1969^[2]William A. Veech constructed examples ofminimal and not uniquely ergodic dynamical systems as follows: "Take two copies of the unit circle and mark off segmentJ of length2πα in the counterclockwise direction on each one with endpoint at 0. Now takeθ irrational and consider the following dynamical system. Start with a pointp, say in the first circle. Rotate counterclockwise by2πθ until the first time the orbit lands inJ; then switch to the corresponding point in the second circle, rotate by2πθ until the first time the point lands inJ; switch back to the first circle and so forth. Veech showed that ifθ is irrational, then there exists irrationalα for which this system is minimal and theLebesgue measure is not uniquely ergodic."^[3]

• Bernoulli map
• Modular arithmetic
• Siegel disc
• Toeplitz algebra
• Phase locking (circle map)
• Weyl sequence

• C. E. Silva,Invitation to ergodic theory, Student Mathematical Library, vol 42,American Mathematical Society, 2008ISBN 978-0-8218-4420-5

• Dynamical systems
• Irrational numbers
• Rotation

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