View source for Topological transitivity

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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====
<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====
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]].