Linear Flow On The Torus

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

:$\mathbb^n = \underbrace_n$

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

:$\frac=\omega_1, \quad \frac=\omega_2,\quad \ldots, \quad \frac=\omega_n.$

The solution of these equations can explicitly be expressed as

:$\Phi_\omega^t(\theta_1, \theta_2, \dots, \theta_n) = (\theta_1+\omega_1 t, \theta_2+\omega_2 t, \dots, \theta_n+\omega_n t) \bmod 2\pi.$

If we represent the torus as $\mathbb = \mathbb^n / \mathbb^n$ 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.

Irrational winding of a torus

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. A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.

Definition

One way of constructing a torus is as the quotient space $\mathbb = \mathbb^2 / \mathbb^2$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection $\pi: \mathbb^2 \to \mathbb$. Each point in the torus has as its preimage one of the translates of the square lattice $\mathbb^2$ in $\mathbb^2$, and $\pi$ factors through a map that takes any point in the plane to a point in the unit square rational, then it can be represented by a fraction and a corresponding lattice point of $\mathbb^2$. It can be shown that then the projection of this line is a simple simple curve">simple closed curve on a torus. If, however, ''k'' is irrational number">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 $\pi$ 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.

Applications

Irrational windings of a torus may be used to set up counter-examples related to dense subspace">dense in the torus.

Applications

Irrational windings of a torus may be used to set up counter-examples related to immersed submanifold but not a Submanifold#Embedded submanifolds">regular submanifold of the torus, which shows that the image of a manifold under a continuous function">continuous