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

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.

Authors and Affiliations

1. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel, e-mail: glasner@math.tau.ac.il, , , , , , IL

   E. Glasner

2. Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel, e-mail: weiss@math.huji.ac.il, , , , , , IL

   B. Weiss

Submitted: October 2001, Revised version: December 2001.

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