View source for Solvable flow

A flow on a [[Solv manifold|solv manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861201.png" /> determined by the action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861202.png" /> of some one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861203.png" /> of the solvable Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861204.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861205.png" /> consists of the cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861206.png" />, then under the action of the solvable flow such a coset goes to the coset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861207.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861208.png" />. A particular case of a solvable flow is a [[Nil flow|nil-flow]]; in the general case the properties of a solvable flow can be considerably more diverse.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Auslander,   L. Green,   F. Hahn,   "Flows on homogeneous spaces" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.M. Stepin,   "Flows on solvmanifolds"  ''Uspekhi Mat. Nauk'' , '''24''' :  5  (1969)  pp. 241  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Auslander,   "An exposition of the structure of solvmanifolds. Part II: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s0861209.png" />-induced flows"  ''Bull. Amer. Math. Soc.'' , '''79''' :  2  (1973)  pp. 262–285</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Safonov,   "Spectral type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612010.png" />-induced ergodic flows"  ''Functional Anal. Appl.'' , '''14''' :  4  (1980)  pp. 315–317  ''Funkts. Anal. i Prilozhen.'' , '''14''' :  4  (1980)  pp. 81–82</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Auslander,   L. Green,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612011.png" />-induced flows and solvmanifolds"  ''Amer. J. Math.'' , '''88'''  (1966)  pp. 43–60</TD></TR></table>



====Comments====
In many cases dynamical properties of the flow, such as [[Ergodicity|ergodicity]], can be deduced from algebraic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612013.png" />. The [[Kronecker theorem|Kronecker theorem]] implies ergodicity for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612015.png" />, the integer lattice, and the flow (written additively) given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612017.png" /> is a coset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612019.png" /> is a fixed vector whose components are linearly independent over the rational numbers. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612021.png" /> is a discrete subgroup, certain one-parameter subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086120/s08612022.png" /> correspond to geodesic and horocycle flow (cf. [[Geodesic flow|Geodesic flow]]; [[Horocycle flow|Horocycle flow]]) on unit tangent bundles of surfaces of constant negative curvature (cf. [[Constant curvature, space of|Constant curvature, space of]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Brezin,   C.C. Moore,   "Flows on homogeneous spaces"  ''Amer. J. Math.'' , '''103'''  (1981)  pp. 571–613</TD></TR></table>