Irrational Cable On A 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 $\left(\theta_1, \theta_2, \ldots, \theta_n\right):$
$\frac = \omega_1, \quad \frac = \omega_2,\quad \ldots, \quad \frac = \omega_n$.

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 $\mathbb^n = \Reals^n / \Z^n$ we see that a starting point is moved by the flow in the direction $\omega = \left(\omega_1, \omega_2, \ldots, \omega_n\right)$ 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, all orbits are either periodic or dense on a subset of the $n$-torus, which is a $k$-torus. When the components of $\omega$ 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 $\omega$ are rationally independent, 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 $\mathbb = \Reals^2 / \Z^2$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection $\pi : \Reals^2 \to \mathbb.$ Each point in the torus has as its preimage one of the translates of the square lattice $\Z^2$ in $\Reals^2,$ and $\pi$ factors through a map that takes any point in the plane to a point in the unit square $$rational, it can be represented by a fraction and a corresponding lattice point of $\Z^2.$ It can be shown that then the projection of this line is a simple curve">simple