Logic Seminar

By a flow, we mean a continuous action of a topological groups $G$ on a compact Hausdorff space $X$. We refer to $X$  as the phase space of the flow. We are primarily interested in minimal flows, that is, flows with no non-trivial proper closed invariant subset. Among minimal flows, there exists a maximal one called the universal minimal flow, $M(G),$ which admits a continuous homomorphism onto every minimal flow. When $G$ is non-Archimedean, that is, it admits a neighbourhood basis of the identity of open subgroups, then $M(G)$ is $0$-dimensional. These are exactly groups of automorphisms of first-order structures with the topology of pointwise convergence. If $M(G)$ is $0$-dimensional, we can think dually in terms of its algebra of clopen subsets.  We summarize which algebras are known to appear as phase spaces of universal minimal flows and we pose questions about the unknown.