Irrational rotation

 ( Dynamical Systems ) 

In the mathematical theory of , an irrational rotation is a map

$T\_\backslash theta\; :\; 0,1\; \backslash rightarrow\; 0,1,\backslash quad\; T\_\backslash theta(x)\; \backslash triangleq\; x\; +\; \backslash theta\; \backslash mod\; 1\; ,$

where is an irrational number. Under the identification of a circle with , or with the interval with the boundary points glued together, this map becomes a rotation of a circle by a proportion of a full revolution (i.e., an angle of  radians). Since is irrational, the rotation has infinite order in the circle group and the map has no .

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

$T\_\backslash theta\; :S^1\; \backslash to\; S^1,\; \backslash quad\; \backslash quad\; \backslash quad\; T\_\backslash theta(x)=xe^\{2\backslash pi\; i\backslash theta\}$

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

$\backslash varphi:(0,1,+)\; \backslash to\; (S^1,\; \backslash cdot)\; \backslash quad\; \backslash varphi(x)=xe^\{2\backslash pi\; i\backslash theta\}$.

It can be shown that is an isometry.

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

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Significance Irrational rotations form a fundamental example in the theory of . According to the Denjoy theorem, every orientation-preserving -diffeomorphism of the circle with an irrational rotation number is topologically conjugate to . 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 is the irrational rotation by . C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

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Properties

• If is irrational, then the orbit of any element of under the rotation is Dense set in . Therefore, irrational rotations are topologically transitive.
• Irrational (and rational) rotations are not topologically mixing.
• Irrational rotations are uniquely ergodicity, with the Lebesgue measure serving as the unique invariant probability measure.
• Suppose . Since is ergodic,
  $\backslash text\{lim\}\; \_\; \{N\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{n=0\}^\{N-1\}\; \backslash chi\_\{[a,b)\}(T\_\backslash theta\; ^n\; (t))=b-a$.

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Generalizations

• Circle rotations are examples of group translations.
• For a general orientation preserving homomorphism of to itself we call a homeomorphism $F:\backslash mathbb\{R\}\backslash to\; \backslash mathbb\{R\}$ a lift of if $\backslash pi\; \backslash circ\; F=f\; \backslash circ\; \backslash pi$ where $\backslash pi\; (t)=t\; \backslash bmod\; 1$.
• 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 an interval exchange transformation.
• Rigid rotations of effectively behave like circle rotations; the invariant measure is the Haar measure.

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Applications

• Skew Products over Rotations of the Circle: In 1969 William A. Veech constructed examples of minimal and not uniquely ergodic dynamical systems as follows: "Take two copies of the unit circle and mark off segment of length in the counterclockwise direction on each one with endpoint at 0. Now take irrational and consider the following dynamical system. Start with a point , say in the first circle. Rotate counterclockwise by until the first time the orbit lands in ; then switch to the corresponding point in the second circle, rotate by until the first time the point lands in ; switch back to the first circle and so forth. Veech showed that if is irrational, then there exists irrational for which this system is minimal and the Lebesgue measure is not uniquely ergodic."

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See also

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

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Further reading

• C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008

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