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 {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 (θ₁, θ₂, ..., θ_n):

{\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \cdots ,\quad {\frac {d\theta _{n}}{dt}}=\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)\mod 2\pi .

If we represent the torus as $\mathbb {T^{n}} =\mathbb {R} ^{n}/\mathbb {Z} ^{n}$ we see that a starting point is moved by the flow in the direction ω = (ω₁, ω₂, ..., 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.^[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 $\mathbb {T^{2}} =\mathbb {R} ^{2}/\mathbb {Z} ^{2}$ of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection $\pi :\mathbb {R} ^{2}\to \mathbb {T^{2}}$ . Each point in the torus has as its preimage one of the translates of the square lattice $\mathbb {Z} ^{2}$ in $\mathbb {R} ^{2}$ , and $\pi$ factors through a map that takes any point in the plane to a point in the unit square $[0,1)^{2}$ given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in $\mathbb {R} ^{2}$ 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 $\mathbb {Z} ^{2}$ . 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 $\pi$ 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 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 $U(1)\times U(1)$ , and the line can be considered as $\mathbb {R}$ . Then it is easy to show that the image of the continuous and analytic group homomorphism $x\mapsto (e^{ix},e^{ikx})$ 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.

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 $\mathbb {R}$ .

References

^ D. P. Zhelobenko (January 1973). Compact Lie groups and their representations. ISBN 9780821886649.
^ ^a ^b ^c Loring W. Tu (2010). An Introduction to Manifolds. Springer. pp. 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

Bibliography

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

[1] D. P. Zhelobenko (January 1973). Compact Lie groups and their representations. ISBN 9780821886649.

[Tu-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

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