The structure of tame minimal dynamical systems for general groups

We use the structure theory of minimal dynamical systems to show that, for a general group Γ , a tame, metric, minimal dynamical system (X, Γ) has the following structure: [Equation not available: see fulltext.]Here (i) X~ is a metric minimal and tame system (ii) η is a strongly proximal extension, (iii) Y is a strongly proximal system, (iv) π is a point distal and RIM extension with unique section, (v) θ, θ^∗ and ι are almost one-to-one extensions, and (vi) σ is an isometric extension. When the map π is also open this diagram reduces to [Equation not available: see fulltext.]In general the presence of the strongly proximal extension η is unavoidable. If the system (X, Γ) admits an invariant measure μ then Y is trivial and X= X~ is an almost automorphic system; i.e. X→ ιZ, where ι is an almost one-to-one extension and Z is equicontinuous. Moreover, μ is unique and ι is a measure theoretical isomorphism ι: (X, μ, Γ) → (Z, λ, Γ) , with λ the Haar measure on Z. Thus, this is always the case when Γ is amenable.