The universal minimal system for the group of homeomorphisms of the Cantor set

Abstract

Each topological group G admits a unique universal minimal dy- namical system (M(G);G). For a locally compact non-compact group this is a nonmetrizable system with a rich structure, on which G acts eectively. 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. In this paper we show that for the topological group G = Homeo(E) of self home- omorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G);G) is isomorphic to Uspenskij's 'max- imal chains' dynamical system (';G) in 22 E . In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the 'dual Ram- sey theorem', a corollary of Graham and Rothschild's Ramsey's theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (';G) and then a general procedure for analyzing G- actions of zero dimensional topological groups is used to show that (M(G);G) is isomorphic to (';G).