4 - Fundamental notions

Summary

Certain important notions in topological dynamics serve as the language, foundation and motivation for the theory. These include pointwise almost periodicity, minimality, distality, proximality, weak-mixing, and topological transitivity for flows. This section is devoted to defining and discussing these concepts and some of their analogs and generalizations to homomorphisms of flows.

Our exposition emphasizes the role played by the semigroups βT, and E(X, T), and the minimal ideals therein in understanding the properties of the flow (X, T). Many of the fundamental notions can be cast in terms of the algebraic structure of these semigroups, their minimal ideals and idempotents. One example of this is proposition 4.9 which shows that (X, T) is distal if and only if E(X, T) is a group. This algebraic approach also leads to 4.12, where it is shown that every distal flow is pointwise almost periodic.

The later part of this section is devoted to a discussion of topological transitivity and related questions. For metric flows, the notions of point transitivity and topological transitivity are quite easily seen to be equivalent (see 4.18). This allows certain results (notably the Furstenberg structure theorem for distal flows) to be deduced for metric flows in a straightforward way. On the other hand topological transitivity and point transitivity are not equivalent in general for flows on compact Hausdorff spaces.