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)