In the mathematical theory of dynamical systems, an irrational rotation is a map T θ : [ 0 , 1 ] → [ 0 , 1 ] , T θ ( x ) ≜ x + θ mod 1 , {\displaystyle...

topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. Denjoy (1932) proved the theorem in the course of his topological...

Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as...

geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences...

F(x)=x+N,} and its rotation number is N {\displaystyle N} (cf. irrational rotation). The rotation number is invariant under topological conjugacy, and even...

SO(3) Rotations and reflections in two dimensions CORDIC Infinitesimal rotation matrix Irrational rotation Orientation (geometry) Rodrigues' rotation formula...

irrational rotation of a circle is ergodic: the orbit of a point is such that eventually, every other point in the circle is visited. Such rotations are...

C*-algebras, the noncommutative tori Aθ, also known as irrational rotation algebras for irrational values of θ, form a family of noncommutative C*-algebras...

the Fatou set where the dynamics is analytically conjugate to an irrational rotation. Given a holomorphic endomorphism f : S → S {\displaystyle f:S\to...

component where the rational function is conformally conjugate to an irrational rotation of the standard annulus. Namely if ƒ possesses a Herman ring U with...

f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. U {\displaystyle U} is a Herman ring:...

time; in particular, if Ω {\displaystyle \Omega } is irrational the map reduces to an irrational rotation. The particular circle map originally studied by...

after the rotation. Four-dimensional rotations are of two types: simple rotations and double rotations. A simple rotation R about a rotation centre O leaves...

an irrational rotation of the open unit disk; or (4) U is a Herman ring, meaning that the action of f on U is conjugate to an irrational rotation of an...

The canonical example of an ergodic system that does not mix is irrational circle rotation. The ergodic decomposition theorem states, roughly, that every...

in the case of zero entropy, if it is Kakutani-equivalent to an irrational rotation of a circle. Shift of finite type Markov chain Hidden Bernoulli model...

attractors. These arise naturally in projective spaces, though classical irrational rotation on the circle can be adapted too. The collection of functions f i...

can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be inaccessible to the techniques...

of angle-preserving transformations Irrational rotation – Rotation of a circle by an angle of π times an irrational number Lens (geometry) – Convex plane...

b/a is irrational, the leaves are noncompact, homeomorphic to the non-compactified real line, and dense in the torus (cf Irrational rotation). The trajectory...

pendulum Dyadic transformation Dynamical system simulation Hénon map Irrational rotation Kaplan–Yorke map List of chaotic maps Lorenz system Quadratic map...

{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x)~.} Irrational rotation Iterated function system Iterative method Rotation number Sarkovskii's theorem Fractional calculus...

called being essentially invariant. An irrational rotation of the circle R/Z, T: x → x + θ, where θ is irrational, is ergodic. This transformation has even...

variation. Suppose that a map f from the circle T to itself has irrational rotation number α, and p/q is a rational approximation to α with p and q coprime...

Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map...

orientations in G(10,4), and the latter has orientations in G(p,4) with the irrational rotation 2π/p = arctan(1/2). They show that G(p,4) is dense in SO(3) for the...

6 (2000), 64–74 V. Afraimovich and T. Young, Relative density of irrational rotation numbers in families of circle diffeomorphisms. Ergodic theory and...

strong 3-mixing. It is known that strong m-mixing implies ergodicity. Irrational rotations of the circle, and more generally irreducible translations on a torus...

counterclockwise rotation of 90 degrees about the origin, which is a quarter of a circle. Multiplication by −i corresponds to a clockwise rotation of 90 degrees...

itself. The measurement of angles is intrinsically linked with circles and rotation. For an ordinary angle, this is often visualized or defined using the arc...