About: Ergodic flow

In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understo