495 (number)

In mathematical theory of dynamical systems, an irrational rotation is a map

$T_{\theta }:[0,1]\to [0,1],T_{\theta }(x)\triangleq x+\theta mod1,$

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 2πθ radians). Since θ is irrational, the rotation has infinite order in the circle group and the map T_θ has no periodic orbits. Moreover, the orbit of any point x under the iterates of T_θ,

$\{x+n\theta :n\in \mathbb{\mathbb{Z}}\},$

is dense in the interval [0, 1) or the circle.

Significance

Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving C²-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to T_θ. 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 θ is the irrational rotation by θ. C*-algebras associated with irrational rotations, known as irrational rotation algebras, 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