Knowlpedia

Example

Kronecker foliation of the two-torus

The foliation by orbits of a constant-slope linear flow, with an irrational rotation as return map.

For irrational θ\theta, the Kronecker foliation of T2\mathbb T^2 has leaves given by the of

Φt(x,y)=(x+t,y+θt) mod Z2.\Phi^t(x,y)=(x+t,y+\theta t)\bmod\mathbb Z^2.

Locally these are parallel straight lines; each leaf is dense. The flow preserves Haar probability.

Rotation on a transversal

The circle {0}×T\{0\}\times\mathbb T meets each leaf. Its first-return map at time one is y↦y+θy\mapsto y+\theta. Two points on this transversal belong to the same leaf exactly when they belong to the same irrational-rotation orbit.

The rotation crossed product is therefore a concrete algebra associated with the transversal dynamics of the leaf space. The full foliation operator algebra and the transversal algebra are related by Morita equivalence; they should not be identified merely by writing the same orbit relation.

References
  1. Alain Connes, Noncommutative Geometry, 1994, Introduction, Kronecker foliation and its transverse rotation.