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:

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

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. $X \overset{\iota }{\rightarrow } Z$, where $\iota$ is an almost one-to-one extension and Z is equicontinuous. Moreover, $\mu$ is unique and $\iota$ is a measure theoretical isomorphism $\iota : (X,\mu , \Gamma ) \rightarrow (Z, \lambda , \Gamma )$, with $\lambda$ the Haar measure on Z. Thus, this is always the case when $\Gamma$ is amenable.