The universal minimal system for the group of homeomorphisms of the Cantor set
Volume 176 / 2003
Abstract
Each topological group $G$ admits a unique universal minimal dynamical system $(M(G),G)$. For a locally compact noncompact group this is a nonmetrizable system with a 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. We show that for the topological group $G=\mathop {\rm Homeo}(E)$ of self-homeomorphisms of the Cantor set $E$, with the topology of uniform convergence, the universal minimal system $(M(G),G)$ is isomorphic to Uspenskij's “maximal chains" dynamical system $({\mit \Phi },G)$ in $2^{2^E}$. In particular it follows that $M(G)$ is homeomorphic to the Cantor set. Our main tool is the “dual Ramsey 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 $({\mit \Phi },G)$, and then a general procedure for analyzing $G$-actions of zero-dimensional topological groups is applied to show that $(M(G),G)$ is isomorphic to $({\mit \Phi },G)$.
