2002/04/10 by Eli Glasner
Mathematics · #math.DS #msc:22A05 #msc:22A10 #msc:54H20
published as Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 119--123, Topology Atlas, Toronto, 2002 · 5 pages. The results in this article will be treated fully in an article, written jointly with B. Weiss, to be published in Geometric and Functional Analysis (GAFA)
arxiv created 2002/04/10 · arxiv updated 2009/11/30
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_∞ of all permutations of the integers \mathbb Z, with the topology of pointwise convergence. We show that (M(S_∞),S_∞) is a symbolic dynamical system (hence in particular M(S_∞) 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_∞) has the structure of a two-to-one group extension of proximal system and that it is uniquely ergodic.