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

which is represented by the following differential equations with respect to the standard angular coordinates

The solution of these equations can explicitly be expressed as

If we represent the torus as we see that a starting point is moved by the flow in the direction at constant speed and when it reaches the border of the unitary -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 -torus which is a -torus. When the components of 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 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 edit

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 edit

One way of constructing a torus is as the quotient space   of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection   Each point in the torus has as its preimage one of the translates of the square lattice   in   and   factors through a map that takes any point in the plane to a point in the unit square   given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in   given by the equation   If the slope   of the line is rational, then it can be represented by a fraction and a corresponding lattice point of   It can be shown that then the projection of this line is a simple closed curve on a torus. If, however,   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   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 edit

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  , and the line can be considered as  . Then it is easy to show that the image of the continuous and analytic group homomorphism   is not a regular submanifold for irrational  [2][3] although it is an immersed submanifold, and therefore a Lie subgroup. It may also be used to show that if a subgroup   of the Lie group   is not closed, the quotient   does not need to be a manifold[4] and might even fail to be a Hausdorff space.

See also edit

Notes edit

^ a: As a topological subspace of the torus, the irrational winding is not a manifold at all, because it is not locally homeomorphic to  .

References edit

  1. ^ D. P. Zhelobenko (January 1973). Compact Lie groups and their representations. ISBN 9780821886649.
  2. ^ a b c 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 edit

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