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A flow on a [[Nil manifold|nil manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666501.png" /> defined by the action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666502.png" /> of some one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666503.png" /> of a nilpotent Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666504.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666505.png" /> consists of the cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666506.png" />, then under the action of the nil flow such a coset at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666507.png" /> goes over in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066650/n0666508.png" />.
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A flow on a [[Nil manifold|nil manifold]] $  M = G / H $
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defined by the action on $  M $
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of some one-parameter subgroup $  g _ {t} $
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of a nilpotent Lie group $  G $:  
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If $  M $
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consists of the cosets $  g H $,  
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then under the action of the nil flow such a coset at time $  t $
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goes over in $  g _ {t} g H $.
  
 
====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></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></table>
 
 
  
 
====Comments====
 
====Comments====
 
The first example of a compact minimal flow that is distal but not equicontinuous was a nil flow (cf. [[Distal dynamical system|Distal dynamical system]]; [[Equicontinuity|Equicontinuity]]).
 
The first example of a compact minimal flow that is distal but not equicontinuous was a nil flow (cf. [[Distal dynamical system|Distal dynamical system]]; [[Equicontinuity|Equicontinuity]]).

Latest revision as of 08:02, 6 June 2020


A flow on a nil manifold $ M = G / H $ defined by the action on $ M $ of some one-parameter subgroup $ g _ {t} $ of a nilpotent Lie group $ G $: If $ M $ consists of the cosets $ g H $, then under the action of the nil flow such a coset at time $ t $ goes over in $ g _ {t} g H $.

References

[1] L. Auslander, L. Green, F. Hahn, "Flows on homogeneous spaces" , Princeton Univ. Press (1963)

Comments

The first example of a compact minimal flow that is distal but not equicontinuous was a nil flow (cf. Distal dynamical system; Equicontinuity).

How to Cite This Entry:
Nil flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_flow&oldid=47972
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article