The structure of tame minimal dynamical systems for general groups

Abstract

We use the structure theory of minimal dynamical systems to show that, for a general group $\Gamma$, a tame, metric, minimal dynamical system $(X, \Gamma)$ has the following structure: \begin{equation*} \xymatrix {& \tilde{X} \ar[dd]_\pi \ar[dl]_\eta & X^* \ar[l]_-{\theta^*} \ar[d]^{\iota} \ar@/^2pc/@{>}^{\pi^*}[dd]\\ X & & Z \ar[d]^\sigma\\ & Y & Y^* \ar[l]^\theta } \end{equation*} Here (i) $\tilde{X}$ is a metric minimal and tame system (ii) $\eta$ is a strongly proximal extension, (iii) $Y$ is a strongly proximal system, (iv) $\pi$ is a point distal and RIM extension with unique section, (v) $\theta$, $\theta^*$ and $\iota$ are almost one-to-one extensions, and (vi) $\sigma$ is an isometric extension. When the map $\pi$ is also open this diagram reduces to \begin{equation*} \xymatrix {& \tilde{X} \ar[dl]_\eta \ar[d]^{\iota} \ar@/^2pc/@{>}^\pi[dd]\\ X & Z \ar[d]^\sigma\\ & Y } \end{equation*} In general the presence of the strongly proximal extension $\eta$ is unavoidable. If the system $(X, \Gamma)$ admits an invariant measure $\mu$ then $Y$ is trivial and $X = \tilde{X}$ is an almost automorphic system; i.e. $X \overset{\iota}{\to} Z$, where $\iota$ is an almost one-to-one extension and $Z$ is equicontinuous. Moreover, $\mu$ is unique and $\iota$ is a measure theoretical isomorphism $\iota : (X,\mu, \Gamma) \to (Z, \lambda, \Gamma)$, with $\lambda$ the Haar measure on $Z$. Thus, this is always the case when $\Gamma$ is amenable.