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

$${\displaystyle \mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}}$$

which is represented by the following differential equations with respect to the standard angular coordinates ${\displaystyle \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):}$

$${\displaystyle {\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \ldots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}.}$$

The solution of these equations can explicitly be expressed as

$${\displaystyle \Phi _{\omega }^{t}(\theta _{1},\theta _{2},\dots ,\theta _{n})=(\theta _{1}+\omega _{1}t,\theta _{2}+\omega _{2}t,\dots ,\theta _{n}+\omega _{n}t){\bmod {2}}\pi .}$$

If we represent the torus as ${\displaystyle \mathbb {T^{n}} =\mathbb {R} ^{n}/\mathbb {Z} ^{n}}$ we see that a starting point is moved by the flow in the direction ${\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)}$ at constant speed and when it reaches the border of the unitary ${\displaystyle n}$-cube it jumps to the opposite face of the cube.

Irrational rotation on a 2-torus

For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the ${\displaystyle n}$-torus which is a ${\displaystyle k}$-torus. When the components of ${\displaystyle \omega }$ 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 ${\displaystyle \omega }$ 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. ^[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.

Definition

One way of constructing a torus is as the quotient space ${\displaystyle \mathbb {T^{2}} =\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection ${\displaystyle \pi$ :\mathbb {R} ^{2}\to \mathbb {T^{2}} .} Each point in the torus has as its preimage one of the translates of the square lattice ${\displaystyle \mathbb {Z} ^{2}}$ in ${\displaystyle \mathbb {R} ^{2},}$ and ${\displaystyle \pi }$ factors through a map that takes any point in the plane to a point in the unit square ${\displaystyle [0,1)^{2}}$ given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in ${\displaystyle \mathbb {R} ^{2}}$ given by the equation ${\displaystyle y=kx.}$ If the slope ${\displaystyle k}$ of the line is rational, then it can be represented by a fraction and a corresponding lattice point of ${\displaystyle \mathbb {Z} ^{2}.}$ It can be shown that then the projection of this line is a simple closed curve on a torus. If, however, ${\displaystyle k}$ 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 ${\displaystyle \pi }$ 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 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 topology of the submanifold does not have to coincide with the subspace topology of the submanifold. ^[2]

Secondly, the torus can be considered as a Lie group ${\displaystyle U(1)\times U(1)}$, and the line can be considered as ${\displaystyle \mathbb {R} }$. Then it is easy to show that the image of the continuous and analytic group homomorphism ${\displaystyle x\mapsto \left(e^{ix},e^{ikx}\right)}$ is not a regular submanifold for irrational ${\displaystyle k,}$ ^{[2] [3]} although it is an immersed submanifold, and therefore a Lie subgroup. It may also be used to show that if a subgroup ${\displaystyle H}$ of the Lie group ${\displaystyle G}$ is not closed, the quotient ${\displaystyle G/H}$ does not need to be a manifold ^[4] and might even fail to be a Hausdorff space.

See also

• Completely integrable system
• Ergodic theory  – Branch of mathematics that studies dynamical systems
• List of topologies  – List of concrete topologies and topological spaces
• Quasiperiodic motion  – Type of chaotic motion
• Torus knot  – Knot which lies on the surface of a torus in 3-dimensional space

Notes

^  a: As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic to ${\displaystyle \mathbb {R} }$.

References

1. D. P. Zhelobenko (January 1973). Compact Lie groups and their representations. ISBN   9780821886649.
2. Loring W. Tu (2010). An Introduction to Manifolds . Springer. pp.  168. ISBN   978-1-4419-7399-3.
3. Čap, Andreas; Slovák, Jan (2009), Parabolic Geometries: Background and general theory, AMS, p. 24, ISBN   978-0-8218-2681-2
4. 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

Bibliography

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