Minimal actions of the group ${\Bbb S(Z)}$ of permutations of the integers

Abstract.

Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact non-compact group this is a nonmetrizable system with a very rich structure, on which G acts effectively. However there are 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 one has an explicit description. One such group is the topological group $\Bbb S$ of all the permutations of the integers $\Bbb Z$, with the topology of pointwise convergence. In this paper we show that (M($\Bbb S$), $\Bbb S$) is a symbolic dynamical system (hence in particular M($\Bbb S$) is a Cantor set), and we give a full description of all its symbolic factors.