The Cantor Set of Linear Orders on N Is the Universal Minimal S∞-system

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 nonmetrizable. 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 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. This is a summary of a talk given at the Prague Topological Symposium of 2001 in which I described results obtained in a joint paper with B. Weiss. The paper is going to appear soon in GAFA [4]. Given a topological group G and a compact Hausdorff space X, a dynamical system (X,G) is a jointly continuous action of G on X. If (Y,G) is a second dynamical system then a continuous onto map π : (X,G) → (Y,G) which intertwines the G actions is called a homomorphism. The dynamical system (X,G) is point transitive if there exists a point x0 ∈ X whose orbit Gx0 is dense in X. (X,G) is minimal if every orbit is dense. It can be easily shown that there exists a unique (up to isomorphism of dynamical 2000 Mathematics Subject Classification. 22A05, 22A10, 54H20.