In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a **linear flow on the torus** is a flow on the *n*-dimensional torus

which is represented by the following differential equations with respect to the standard angular coordinates (*θ*_{1}, *θ*_{2}, ..., *θ*_{n}):

The solution of these equations can explicitly be expressed as

If we represent the torus as we see that a starting point is moved by the flow in the direction *ω* = (*ω*_{1}, *ω*_{2}, ..., *ω*_{n}) at constant speed and when it reaches the border of the unitary *n*-cube it jumps to the opposite face of the cube.

For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the *n*-torus which is a *k*-torus. When the components of ω are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two dimensional case: if the two components of ω are rationally independent then the 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 a consequence the orbits of the flow must be dense on the torus.

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## Irrational winding of a torusEdit

In topology, an **irrational winding of a torus** is a continuous injection of a line into a two-dimensional torus that is used to set up several counterexamples.^{[1]} A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.

### DefinitionEdit

One way of constructing a torus is as the quotient space of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection . Each point in the torus has as its preimage one of the translates of the square lattice in , and factors through a map that takes any point in the plane to a point in the unit square given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in given by the equation *y = kx*. If the slope *k* of the line is rational, then it can be represented by a fraction and a corresponding lattice point of . It can be shown that then the projection of this line is a simple closed curve on a torus. If, however, *k* is irrational, then it will not cross any lattice points except 0, which means that its projection on the torus will not be a closed curve, and the restriction of on this line is injective. Moreover, it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, is dense in the torus.

### ApplicationsEdit

Irrational windings of a torus may be used to set up counter-examples related to monomorphisms. An irrational winding is an immersed submanifold but not a regular submanifold of the torus, which shows that the image of a manifold under a continuous injection to another manifold is not necessarily a (regular) submanifold.^{[2]} Irrational windings are also examples of the fact that the induced submanifold topology does not have to coincide with the subspace topology of the submanifold.^{[2]}

Secondly, the torus can be considered as a Lie group , and the line can be considered as . Then it is easy to show that the image of the continuous and analytic group homomorphism is not a regular submanifold for irrational k,^{[2]}^{[3]} although it is an immersed submanifold, and therefore a Lie subgroup. It may also be used to show that if a subgroup *H* of the Lie group *G* is not closed, the quotient *G*/*H* does not need to be a manifold^{[4]} and might even fail to be a Hausdorff space.

## See alsoEdit

## NotesEdit

**^** **a:** As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic to .

## ReferencesEdit

**^**D. P. Zhelobenko.*Compact Lie groups and their representations*.- ^
^{a}^{b}^{c}Loring W. Tu (2010).*An Introduction to Manifolds*. Springer. p. 168. ISBN 978-1-4419-7399-3. **^**Čap, Andreas; Slovák, Jan (2009),*Parabolic Geometries: Background and general theory*, AMS, p. 24, ISBN 978-0-8218-2681-2**^**Sharpe, R.W. (1997),*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*, Springer-Verlag, New York, p. 146, ISBN 0-387-94732-9

## BibliographyEdit

- Anatole Katok and Boris Hasselblatt (1996).
*Introduction to the modern theory of dynamical systems*. Cambridge. ISBN 0-521-57557-5.