The Cantor set of linear orders on N is the universal minimal S_\infty-system

Abstract

Each topological group $G$ admits a unique universal minimal dynamical system $(M(G),G)$. When $G$ is a non-compact locally compact group the phase space $M(G)$ of this universal system is non-metrizable. There are however topological groups for which $M(G)$ is the trivial one point system (extremely amenable groups), as well as topological groups $G$ for which $M(G)$ is a metrizable space and for which there is an explicit description of the dynamical system $(M(G),G)$. One such group is the topological group $S_\infty$ of all permutations of the integers ${\mathbb Z}$, with the topology of pointwise convergence. We show that $(M(S_\infty),S_\infty)$ is a symbolic dynamical system (hence in particular $M(S_\infty)$ is a Cantor set), and give a full description of all its symbolic factors. Among other facts we show that $(M(G),G)$ (and hence also every minimal $S_\infty$) has the structure of a two-to-one group extension of proximal system and that it is uniquely ergodic.