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A property defined for a [[Topological dynamical system|topological dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931701.png" />, usually for a flow or a cascade (the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931702.png" /> runs through the real numbers or the integers). It consists of the existence of a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931703.png" /> that has the whole phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931704.png" /> as its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931706.png" />-limit set. (Cf. [[Limit set of a trajectory|Limit set of a trajectory]]; an equivalent condition is the existence of a positive semi-trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931707.png" /> that is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931708.png" />.) Such a trajectory (semi-trajectory) is called topologically transitive.
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Closely related to topological transitivity is the property of transitivity of domains: For any non-empty open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t0931709.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317011.png" />. More precisely, topological transitivity implies transitivity of domains, and the converse holds (cf. [[#References|[1]]], [[#References|[2]]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317012.png" /> is a complete separable metric space (in this case the set of topologically-transitive trajectories has the cardinality of the continuum). Hence, with the same hypotheses on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317013.png" />, the property of topological transitivity is symmetric with respect to the time direction: If there exists a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317014.png" /> having the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317015.png" /> as its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317017.png" />-limit set, then one has transitivity of domains and topological transitivity.
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Often topological transitivity is used to mean the existence of a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317018.png" /> that is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317019.png" />. (The difference between the definitions is essential when the points of this trajectory form an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317020.png" />; otherwise it is itself an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317021.png" />-limit or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317022.png" />-limit, and hence the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317023.png" /> is its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317024.png" />-limit or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093170/t09317025.png" />-limit set.) The last definition is also used for more general transformation groups [[#References|[3]]]. The definition and some of the results also carry over to the case of non-invertible mappings and semi-groups, although one is usually not concerned with these in topological dynamics.
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A property defined for a [[Topological dynamical system|topological dynamical system]]  $  \{ T _ {t} \} $,
 +
usually for a flow or a cascade (the time  $  t $
 +
runs through the real numbers or the integers). It consists of the existence of a trajectory  $  \{ T _ {t} w _ {0} \} $
 +
that has the whole phase space  $  W $
 +
as its  $  \omega $-
 +
limit set. (Cf. [[Limit set of a trajectory|Limit set of a trajectory]]; an equivalent condition is the existence of a positive semi-trajectory $  \{ {T _ {t} w _ {0} } : {t \geq  0 } \} $
 +
that is everywhere dense in $  W $.)
 +
Such a trajectory (semi-trajectory) is called topologically transitive.
 +
 
 +
Closely related to topological transitivity is the property of transitivity of domains: For any non-empty open sets  $  U, V \subset  W $
 +
there is a  $  t > 0 $
 +
such that  $  T _ {t} U \cap V \neq \emptyset $.
 +
More precisely, topological transitivity implies transitivity of domains, and the converse holds (cf. [[#References|[1]]], [[#References|[2]]]) if  $  W $
 +
is a complete separable metric space (in this case the set of topologically-transitive trajectories has the cardinality of the continuum). Hence, with the same hypotheses on  $  W $,
 +
the property of topological transitivity is symmetric with respect to the time direction: If there exists a trajectory  $  \{ T _ {t} w _ {0} \} $
 +
having the whole of  $  W $
 +
as its  $  \alpha $-
 +
limit set, then one has transitivity of domains and topological transitivity.
 +
 
 +
Often topological transitivity is used to mean the existence of a trajectory  $  \{ T _ {t} w _ {0} \} $
 +
that is everywhere dense in  $  W $.  
 +
(The difference between the definitions is essential when the points of this trajectory form an open set in $  W $;  
 +
otherwise it is itself an $  \alpha $-
 +
limit or $  \omega $-
 +
limit, and hence the whole of $  W $
 +
is its $  \alpha $-
 +
limit or $  \omega $-
 +
limit set.) The last definition is also used for more general transformation groups [[#References|[3]]]. The definition and some of the results also carry over to the case of non-invertible mappings and semi-groups, although one is usually not concerned with these in topological dynamics.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.D. Birkhoff,  "Dynamical systems" , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.H. Gottschalk,  G.A. Hedlund,  "Topological dynamics" , Amer. Math. Soc.  (1955)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.D. Birkhoff,  "Dynamical systems" , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.H. Gottschalk,  G.A. Hedlund,  "Topological dynamics" , Amer. Math. Soc.  (1955)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A topological dynamical system with the property of transitivity of domains is also called topological ergodic (in [[#References|[3]]] it is called regionally transitive). In many cases, topological transitivity is implied by [[Metric transitivity|metric transitivity]].
 
A topological dynamical system with the property of transitivity of domains is also called topological ergodic (in [[#References|[3]]] it is called regionally transitive). In many cases, topological transitivity is implied by [[Metric transitivity|metric transitivity]].

Latest revision as of 08:25, 6 June 2020


A property defined for a topological dynamical system $ \{ T _ {t} \} $, usually for a flow or a cascade (the time $ t $ runs through the real numbers or the integers). It consists of the existence of a trajectory $ \{ T _ {t} w _ {0} \} $ that has the whole phase space $ W $ as its $ \omega $- limit set. (Cf. Limit set of a trajectory; an equivalent condition is the existence of a positive semi-trajectory $ \{ {T _ {t} w _ {0} } : {t \geq 0 } \} $ that is everywhere dense in $ W $.) Such a trajectory (semi-trajectory) is called topologically transitive.

Closely related to topological transitivity is the property of transitivity of domains: For any non-empty open sets $ U, V \subset W $ there is a $ t > 0 $ such that $ T _ {t} U \cap V \neq \emptyset $. More precisely, topological transitivity implies transitivity of domains, and the converse holds (cf. [1], [2]) if $ W $ is a complete separable metric space (in this case the set of topologically-transitive trajectories has the cardinality of the continuum). Hence, with the same hypotheses on $ W $, the property of topological transitivity is symmetric with respect to the time direction: If there exists a trajectory $ \{ T _ {t} w _ {0} \} $ having the whole of $ W $ as its $ \alpha $- limit set, then one has transitivity of domains and topological transitivity.

Often topological transitivity is used to mean the existence of a trajectory $ \{ T _ {t} w _ {0} \} $ that is everywhere dense in $ W $. (The difference between the definitions is essential when the points of this trajectory form an open set in $ W $; otherwise it is itself an $ \alpha $- limit or $ \omega $- limit, and hence the whole of $ W $ is its $ \alpha $- limit or $ \omega $- limit set.) The last definition is also used for more general transformation groups [3]. The definition and some of the results also carry over to the case of non-invertible mappings and semi-groups, although one is usually not concerned with these in topological dynamics.

References

[1] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
[2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[3] W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)

Comments

A topological dynamical system with the property of transitivity of domains is also called topological ergodic (in [3] it is called regionally transitive). In many cases, topological transitivity is implied by metric transitivity.

How to Cite This Entry:
Topological transitivity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_transitivity&oldid=13191
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article