Mathematics – Dynamical Systems
Scientific paper
2002-04-10
Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 119--123, Topology Atlas, Toronto, 2002
Mathematics
Dynamical Systems
5 pages. The results in this article will be treated fully in an article, written jointly with B. Weiss, to be published in Ge
Scientific paper
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.
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