Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map
where θ is an
periodic orbits
.
Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map
The relationship between the additive and multiplicative notations is the group isomorphism
- .
It can be shown that φ is an isometry.
There is a strong distinction in circle rotations that depends on whether θ is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if and , then when . It can also be shown that when .
Significance
Irrational rotations form a fundamental example in the theory of
ergodic transformation, but it is not mixing. The Poincaré map for the dynamical system associated with the Kronecker foliation on a torus with angle θ> is the irrational rotation by θ. C*-algebras associated with irrational rotations, known as irrational rotation algebras
, have been extensively studied.
Properties
- If θ is irrational, then the orbit of any element of [0, 1] under the rotation Tθ is topologically transitive.
- Irrational (and rational) rotations are not topologically mixing.
- Irrational rotations are uniquely ergodic, with the Lebesgue measure serving as the unique invariant probability measure.
- Suppose [a, b] ⊂ [0, 1]. Since Tθ is ergodic,
.
Generalizations
- Circle rotations are examples of group translations.
- For a general orientation preserving homomorphism f of S1 to itself we call a homeomorphism a lift of f if where .[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 compact groups effectively behave like circle rotations; the invariant measure is the Haar measure.
Applications
- Skew Products over Rotations of the Circle: In 1969[2] 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 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 Lebesgue measure is not uniquely ergodic."[3]
See also
- Bernoulli map
- Modular arithmetic
- Siegel disc
- Toeplitz algebra
- Phase locking(circle map)
- Weyl sequence
References
- ^ Fisher, Todd (2007). "Circle Homomorphisms" (PDF).
- ^
PMID 16591677.
- ^ Masur, Howard; Tabachnikov, Serge (2002). "Rational Billiards and Flat Structures". In Hasselblatt, B.; Katok, A. (eds.). Handbook of Dynamical Systems (PDF). Vol. IA. Elsevier.
Further reading
- C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, ISBN 978-0-8218-4420-5