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A dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334501.png" /> with a metric phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334502.png" /> such that for any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334503.png" /> the greatest lower bound of the distances,
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A dynamical system $\{T^t\}$ with a metric phase space $X$ such that for any points $x\neq y$ the greatest lower bound of the distances,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334504.png" /></td> </tr></table>
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$$\inf_t\rho(T^tx,T^ty)>0.$$
  
If a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334505.png" /> in a given dynamical system has this property, one says that this pair of points is distal; thus, a distal dynamical system is a dynamical system all pairs of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334506.png" /> of which are distal.
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If a pair of points $x\neq y$ in a given dynamical system has this property, one says that this pair of points is distal; thus, a distal dynamical system is a dynamical system all pairs of points $x\neq y$ of which are distal.
  
This definition is suitable for  "general"  dynamical systems, when the  "time"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334507.png" /> runs through an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334508.png" />. Interesting results are obtained if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d0334509.png" /> is locally compact (the  "classical"  cases of a cascade or flow, viz. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345011.png" />, are fundamental, but their treatment is hardly simpler), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345012.png" /> is compact. Of special interest is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345013.png" /> is a [[Minimal set|minimal set]] (the general case is reducible, in a sense, to this case, since under the above restriction (the closure of) each trajectory is a minimal set). The most important example of a distal dynamical system is the system resulting from the closure of an almost-periodic trajectory of some dynamical system. A second example are nil-flows [[#References|[1]]]. As is also the case in the above examples, the construction of a distal dynamical system with a minimal set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345014.png" /> under these conditions permits a fairly detailed description of an algebraic nature [[#References|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#References|[3]]].
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This definition is suitable for  "general"  dynamical systems, when the  "time"  $t$ runs through an arbitrary group $G$. Interesting results are obtained if $G$ is locally compact (the  "classical"  cases of a cascade or flow, viz. when $G=\mathbf Z$ or $G=\mathbf R$, are fundamental, but their treatment is hardly simpler), and $X$ is compact. Of special interest is the case when $X$ is a [[Minimal set|minimal set]] (the general case is reducible, in a sense, to this case, since under the above restriction (the closure of) each trajectory is a minimal set). The most important example of a distal dynamical system is the system resulting from the closure of an almost-periodic trajectory of some dynamical system. A second example are nil-flows [[#References|[1]]]. As is also the case in the above examples, the construction of a distal dynamical system with a minimal set $X$ under these conditions permits a fairly detailed description of an algebraic nature [[#References|[2]]]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [[#References|[3]]].
  
 
====References====
 
====References====
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There are several notions of  "almost-periodic trajectory"  in use. In the article above, an almost-periodic trajectory of a point in a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] is a trajectory such that the flow is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]) on the orbit closure of this point (cf. [[#References|[a7]]]; such a trajectory is also called uniformly almost-periodic, [[#References|[3]]]).
 
There are several notions of  "almost-periodic trajectory"  in use. In the article above, an almost-periodic trajectory of a point in a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] is a trajectory such that the flow is equicontinuous (cf. [[Equicontinuity|Equicontinuity]]) on the orbit closure of this point (cf. [[#References|[a7]]]; such a trajectory is also called uniformly almost-periodic, [[#References|[3]]]).
  
The Furstenberg structure theorem referred to above was proved originally for any distal minimal dynamical system on a compact metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345015.png" /> and arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345016.png" />. In [[#References|[a2]]], the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345017.png" /> is metrizable was removed. There is also a so-called  "relative"  version of Furstenberg's theorem, applicable to distal morphisms between compact minimal dynamical systems: see [[#References|[a1]]], (15.4) or [[#References|[3]]], (3.14.22) for the case that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345018.png" /> is metrizable or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345019.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033450/d03345020.png" />-compact (see also [[#References|[a4]]]) and [[#References|[a6]]] for the general case. For yet further generalizations (e.g., to point-distal morphisms, the so-called Veech structure theorem) see [[#References|[3]]], (3.15.42), [[#References|[a3]]] and [[#References|[a5]]].
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The Furstenberg structure theorem referred to above was proved originally for any distal minimal dynamical system on a compact metric space $X$ and arbitrary group $G$. In [[#References|[a2]]], the condition that $X$ is metrizable was removed. There is also a so-called  "relative"  version of Furstenberg's theorem, applicable to distal morphisms between compact minimal dynamical systems: see [[#References|[a1]]], (15.4) or [[#References|[3]]], (3.14.22) for the case that either $X$ is metrizable or $G$ is $\sigma$-compact (see also [[#References|[a4]]]) and [[#References|[a6]]] for the general case. For yet further generalizations (e.g., to point-distal morphisms, the so-called Veech structure theorem) see [[#References|[3]]], (3.15.42), [[#References|[a3]]] and [[#References|[a5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Ellis,  "Lectures on topological dynamics" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Ellis,  "The Furstenberg structure theorem"  ''Pacific J. Math.'' , '''76'''  (1978)  pp. 345–349</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Ellis,  "The Veech structure theorem"  ''Trans. Amer. Math. Soc.'' , '''186'''  (1973)  pp. 203–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Ihrig,  D. McMahon,  "On distal flows of finite codimension"  ''Indian Univ. Math. J.'' , '''33'''  (1984)  pp. 345–351</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. McMahon,  L.J. Nachman,  "An instrinsic characterization for PI-flows"  ''Pacific J. Math.'' , '''89'''  (1980)  pp. 391–403</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. McMahon,  T.S. Wu,  "Distal homomorphisms of non-metric minimal flows"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 283–287</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Ellis,  "Lectures on topological dynamics" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Ellis,  "The Furstenberg structure theorem"  ''Pacific J. Math.'' , '''76'''  (1978)  pp. 345–349</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Ellis,  "The Veech structure theorem"  ''Trans. Amer. Math. Soc.'' , '''186'''  (1973)  pp. 203–218</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Ihrig,  D. McMahon,  "On distal flows of finite codimension"  ''Indian Univ. Math. J.'' , '''33'''  (1984)  pp. 345–351</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. McMahon,  L.J. Nachman,  "An instrinsic characterization for PI-flows"  ''Pacific J. Math.'' , '''89'''  (1980)  pp. 391–403</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. McMahon,  T.S. Wu,  "Distal homomorphisms of non-metric minimal flows"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 283–287</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR></table>

Latest revision as of 11:55, 26 July 2014

A dynamical system $\{T^t\}$ with a metric phase space $X$ such that for any points $x\neq y$ the greatest lower bound of the distances,

$$\inf_t\rho(T^tx,T^ty)>0.$$

If a pair of points $x\neq y$ in a given dynamical system has this property, one says that this pair of points is distal; thus, a distal dynamical system is a dynamical system all pairs of points $x\neq y$ of which are distal.

This definition is suitable for "general" dynamical systems, when the "time" $t$ runs through an arbitrary group $G$. Interesting results are obtained if $G$ is locally compact (the "classical" cases of a cascade or flow, viz. when $G=\mathbf Z$ or $G=\mathbf R$, are fundamental, but their treatment is hardly simpler), and $X$ is compact. Of special interest is the case when $X$ is a minimal set (the general case is reducible, in a sense, to this case, since under the above restriction (the closure of) each trajectory is a minimal set). The most important example of a distal dynamical system is the system resulting from the closure of an almost-periodic trajectory of some dynamical system. A second example are nil-flows [1]. As is also the case in the above examples, the construction of a distal dynamical system with a minimal set $X$ under these conditions permits a fairly detailed description of an algebraic nature [2]; for an account of the theory of distal dynamic systems and their generalizations, as well as the relevant literature, see [3].

References

[1] L. Auslander, L. Green, F. Hahn, "Flows on homogeneous spaces" , Princeton Univ. Press (1963)
[2] H. Furstenberg, "The structure of distal flows" Amer. J. Math. , 85 : 3 (1963) pp. 477–515
[3] I.U. Bronshtein, "Extensions of minimal transformation groups" , Sijthoff & Noordhoff (1979) (Translated from Russian)


Comments

There are several notions of "almost-periodic trajectory" in use. In the article above, an almost-periodic trajectory of a point in a flow (continuous-time dynamical system) is a trajectory such that the flow is equicontinuous (cf. Equicontinuity) on the orbit closure of this point (cf. [a7]; such a trajectory is also called uniformly almost-periodic, [3]).

The Furstenberg structure theorem referred to above was proved originally for any distal minimal dynamical system on a compact metric space $X$ and arbitrary group $G$. In [a2], the condition that $X$ is metrizable was removed. There is also a so-called "relative" version of Furstenberg's theorem, applicable to distal morphisms between compact minimal dynamical systems: see [a1], (15.4) or [3], (3.14.22) for the case that either $X$ is metrizable or $G$ is $\sigma$-compact (see also [a4]) and [a6] for the general case. For yet further generalizations (e.g., to point-distal morphisms, the so-called Veech structure theorem) see [3], (3.15.42), [a3] and [a5].

References

[a1] R. Ellis, "Lectures on topological dynamics" , Benjamin (1969)
[a2] R. Ellis, "The Furstenberg structure theorem" Pacific J. Math. , 76 (1978) pp. 345–349
[a3] R. Ellis, "The Veech structure theorem" Trans. Amer. Math. Soc. , 186 (1973) pp. 203–218
[a4] E. Ihrig, D. McMahon, "On distal flows of finite codimension" Indian Univ. Math. J. , 33 (1984) pp. 345–351
[a5] D. McMahon, L.J. Nachman, "An instrinsic characterization for PI-flows" Pacific J. Math. , 89 (1980) pp. 391–403
[a6] D. McMahon, T.S. Wu, "Distal homomorphisms of non-metric minimal flows" Proc. Amer. Math. Soc. , 82 (1981) pp. 283–287
[a7] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
How to Cite This Entry:
Distal dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distal_dynamical_system&oldid=15966
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article