View source for Topological transitivity

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.

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.

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.

====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]].