The structure of tame minimal dynamical systems for general groups

• 成果类型: Article
• 署名作者: Glasner, Eli
• 署名单位: Tel Aviv University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0747-z
• 发表日期: 2018
• 页码: 213-244
• 关键词: STRUCTURE THEOREM; distal flows; EXTENSIONS; pairs

摘要

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: [GRAPHICS] Here (i) (X) over tilde 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 tau are almost one-to-one extensions, and (vi) sigma is an isometric extension. When the map pi is also open this diagram reduces to [GRAPHICS] In general the presence of the strongly proximal extension. is unavoidable. If the system (X, Gamma) admits an invariant measure mu then Y is trivial and X = (X) over tilde is an almost automorphic system; i.e. X ->(iota) Z, where iota is an almost one-to-one extension and Z is equicontinuous. Moreover, mu is unique and. is a measure theoretical isomorphism iota : (X, mu, Gamma) -> ( Z, lambda, Gamma), with lambda the Haar measure on Z. Thus, this is always the case when Gamma is amenable.