TRANSITIVE FLOWS, ASSOCIATED CONGRUENCES AND GROUPS
Flows in topological dynamics are essentially actions of semigroups and find their earlier roots in qualitative theory of differential equations. Over time the abstract theory of flows has evolved on its own and pioneering works of R. Ellis, Gottchalk er al have significantlly advanced the theoty since. Particularly, transitive flows ( i.e. those which contain a dense orbit ) were much investigated. To this end, the approach in classification and representation of these flows in the earlier works considered discrete actions and the Banach algebra of bounded complex valued continuous functions on the Stone-Czech compactification of the acting semigroup to realize the representation via subalgebras.
In this talk we will consider an alternate (semigroup) approach and realize transitive flows as quotients of a universal flow via closed congruences. Utilizing this intrinsic approach we will also show the extent to which the associated left congruences determine important properties of minimality, distality and proximality of flows.
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