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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.
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A flow on a [[Solv manifold|solv manifold]] $  M = G/H $
 +
determined by the action on $  M $
 +
of some one-parameter subgroup $  g _ {t} $
 +
of the solvable Lie group $  G $:  
 +
If $  M $
 +
consists of the cosets $  gH $,  
 +
then under the action of the solvable flow such a coset goes to the coset $  g _ {t} gH $
 +
at time $  t $.  
 +
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====
 
====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>
 
<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====
 
====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]]).
+
In many cases dynamical properties of the flow, such as [[Ergodicity|ergodicity]], can be deduced from algebraic properties of $  G $
 +
and $  H $.  
 +
The [[Kronecker theorem|Kronecker theorem]] implies ergodicity for the case $  G = \mathbf R  ^ {n} $,  
 +
$  H = \mathbf Z  ^ {n} $,  
 +
the integer lattice, and the flow (written additively) given by $  g _ {t} ( x + \mathbf Z  ^ {n} ) = x + t a + \mathbf Z  ^ {n} $,  
 +
where $  x + \mathbf Z  ^ {n} $
 +
is a coset of $  \mathbf R  ^ {n} / \mathbf Z  ^ {n} $
 +
and $  a \in \mathbf R  ^ {n} $
 +
is a fixed vector whose components are linearly independent over the rational numbers. When $  G= \mathop{\rm SL} ( 2, \mathbf R ) $
 +
and $  H $
 +
is a discrete subgroup, certain one-parameter subgroups of $  G $
 +
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====
 
====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>
 
<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>

Latest revision as of 08:14, 6 June 2020


A flow on a solv manifold $ M = G/H $ determined by the action on $ M $ of some one-parameter subgroup $ g _ {t} $ of the solvable Lie group $ G $: If $ M $ consists of the cosets $ gH $, then under the action of the solvable flow such a coset goes to the coset $ g _ {t} gH $ at time $ t $. 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: -induced flows" Bull. Amer. Math. Soc. , 79 : 2 (1973) pp. 262–285
[4] A.V. Safonov, "Spectral type of -induced ergodic flows" Functional Anal. Appl. , 14 : 4 (1980) pp. 315–317 Funkts. Anal. i Prilozhen. , 14 : 4 (1980) pp. 81–82
[5] L. Auslander, L. Green, "-induced flows and solvmanifolds" Amer. J. Math. , 88 (1966) pp. 43–60

Comments

In many cases dynamical properties of the flow, such as ergodicity, can be deduced from algebraic properties of $ G $ and $ H $. The Kronecker theorem implies ergodicity for the case $ G = \mathbf R ^ {n} $, $ H = \mathbf Z ^ {n} $, the integer lattice, and the flow (written additively) given by $ g _ {t} ( x + \mathbf Z ^ {n} ) = x + t a + \mathbf Z ^ {n} $, where $ x + \mathbf Z ^ {n} $ is a coset of $ \mathbf R ^ {n} / \mathbf Z ^ {n} $ and $ a \in \mathbf R ^ {n} $ is a fixed vector whose components are linearly independent over the rational numbers. When $ G= \mathop{\rm SL} ( 2, \mathbf R ) $ and $ H $ is a discrete subgroup, certain one-parameter subgroups of $ G $ 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
How to Cite This Entry:
Solvable flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solvable_flow&oldid=14625
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article