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 $${\displaystyle \mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}}$${\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 \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):} $${\displaystyle {\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \ldots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}}$${\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 $${\displaystyle \Phi _{\omega }^{t}(\theta _{1},\theta _{2},\dots ,\theta _{n})=(\theta _{1}+\omega _{1}t,\theta _{2}+\omega _{2}t,\dots ,\theta _{n}+\omega _{n}t){\bmod {2}}\pi }$${\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 ${\displaystyle \mathbb {T} ^{n}=\mathbb {R} ^{n}/\mathbb {Z} ^{n}}${\displaystyle \mathbb {T} ^{n}=\mathbb {R} ^{n}/\mathbb {Z} ^{n}} we see that a starting point is moved by the flow in the direction ${\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)}${\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)} at constant speed and when it reaches the border of the unitary ${\displaystyle n}${\displaystyle 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 ${\displaystyle n}${\displaystyle n}-torus, which is a ${\displaystyle k}${\displaystyle k}-torus. When the components of ${\displaystyle \omega }${\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 ${\displaystyle \omega }${\displaystyle \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.

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.

One way of constructing a torus is as the quotient space ${\displaystyle \mathbb {T^{2}} =\mathbb {R} ^{2}/\mathbb {Z} ^{2}}${\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 ${\displaystyle \pi :\mathbb {R} ^{2}\to \mathbb {T^{2}} .}${\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 ${\displaystyle \mathbb {Z} ^{2}}${\displaystyle \mathbb {Z} ^{2}} in ${\displaystyle \mathbb {R} ^{2},}${\displaystyle \mathbb {R} ^{2},} and ${\displaystyle \pi }${\displaystyle \pi } factors through a map that takes any point in the plane to a point in the unit square ${\displaystyle [0,1)^{2}}${\displaystyle [0,1)^{2}} given by the fractional parts of the original point's Cartesian coordinates.

Now consider a line in ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}} given by the equation ${\displaystyle y=kx.}${\displaystyle y=kx.} If the slope ${\displaystyle k}${\displaystyle k} of the line is rational, it can be represented by a fraction and a corresponding lattice point of ${\displaystyle \mathbb {Z} ^{2}.}${\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, ${\displaystyle k}${\displaystyle k} is irrational, 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 ${\displaystyle \pi }${\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.

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

• Completely integrable system – Property of certain dynamical systemsPages displaying short descriptions of redirect targets
• Ergodic theory – Branch of mathematics that studies dynamical systems
• List of topologies – List of concrete topologies and topological spaces
• Quasiperiodic motion – Type of motion that is approximately periodic
• Torus knot – Knot which lies on the surface of a torus in 3-dimensional space

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

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

• General topology
• Lie groups
• Topological spaces
• Dynamical systems
• Ergodic theory

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