Topological transitivity

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.