Solvable flow
A flow on a solv manifold determined by the action on
of some one-parameter subgroup
of the solvable Lie group
: If
consists of the cosets
, then under the action of the solvable flow such a coset goes to the coset
at time
. A particular case of a solvable flow is a nil-flow; in the general case the properties of a solvable flow can be considerably more diverse.
References
[1] | L. Auslander, L. Green, F. Hahn, "Flows on homogeneous spaces" , Princeton Univ. Press (1963) |
[2] | A.M. Stepin, "Flows on solvmanifolds" Uspekhi Mat. Nauk , 24 : 5 (1969) pp. 241 (In Russian) |
[3] | L. Auslander, "An exposition of the structure of solvmanifolds. Part II: ![]() |
[4] | A.V. Safonov, "Spectral type of ![]() |
[5] | L. Auslander, L. Green, "![]() |
Comments
In many cases dynamical properties of the flow, such as ergodicity, can be deduced from algebraic properties of and
. The Kronecker theorem implies ergodicity for the case
,
, the integer lattice, and the flow (written additively) given by
, where
is a coset of
and
is a fixed vector whose components are linearly independent over the rational numbers. When
and
is a discrete subgroup, certain one-parameter subgroups of
correspond to geodesic and horocycle flow (cf. Geodesic flow; Horocycle flow) on unit tangent bundles of surfaces of constant negative curvature (cf. Constant curvature, space of).
References
[a1] | J. Brezin, C.C. Moore, "Flows on homogeneous spaces" Amer. J. Math. , 103 (1981) pp. 571–613 |
Solvable flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solvable_flow&oldid=48748