TY - JOUR

T1 - The structure of tame minimal dynamical systems for general groups

AU - Glasner, Eli

N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - 54H20

UR - http://www.scopus.com/inward/record.url?scp=85024474093&partnerID=8YFLogxK

U2 - 10.1007/s00222-017-0747-z

DO - 10.1007/s00222-017-0747-z

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AN - SCOPUS:85024474093

SN - 0020-9910

VL - 211

SP - 213

EP - 244

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -