The Cantor set of linear orders in \(\mathbb{N}\) is the universal minimal \(S_\infty\)-system (Q2782211)

For any topological group \(G\), there is an associated (unique up to isomorphism of topological dynamical systems) universal minimal dynamical system \((M(G),G)\). This is universal in the sense that any minimal system \((X,G)\) is a factor of \((M(G),G)\). When \(G\) is non-compact but locally compact, \(M(G)\) is necessarily non-metrizable. On the other hand, there are \textsl{extremely amenable} groups for which \(M(G)\) is a singleton. In this note -- a summary of a talk given at the Prague Topological Symposium -- recent results of \textit{E. Glasner} and \textit{B. Weiss} [Geom. Funct. Anal., to appear (2002)] are stated. These show that the group \(S=S_{\infty}\) of all permutations of a countable set lies between the two extremes: \(M(S)\) is a Cantor set, and a symbolic description of \((M(S),S)\) is given. In addition, it is shown that \((M(S),S)\) is a two-point group extension of a proximal system and that it is uniquely ergodic.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].