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]_π\ar[dl]_η& X^* \ar[l]_-{θ^*} \ar[d]^ι \ar@/^2pc/@{>}^{π^*}[dd]\\ X & & Z \ar[d]^σ\\ & Y & Y^* \ar[l]^θ} \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) $\theta$, ${\theta }^{\ast }$ 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]_η\ar[d]^ι \ar@/^2pc/@{>}^π[dd]\\ X & Z \ar[d]^σ\\ & 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. $X\stackrel{\iota }{\to }Z$, 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.

27 pages; to appear in Invent. Math. arXiv admin note: substantial text overlap with arXiv:math/0609503

References in corpus (1)

• The structure of tame minimal dynamical systems

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