The structure of tame minimal dynamical systems
The structure of tame minimal dynamical systems
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DOI：
10.1017/s0143385707000296
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发表时间：
2006-09
影响因子：
0.9
通讯作者：
E. Glasner
中科院分区：
文献类型：
--
作者：
E. Glasner
Bourgain-Fremlin-Talagrand二分法的动力学版本表明，动力系统的包络半群要么非常大，并包含$\beta \mathbb {N}$的拓扑副本，要么它是一个拓扑由序列的收敛性决定的“驯服”拓扑空间。在后一种情况下，动力系统被称为驯服的。本文利用极小动力系统的结构理论证明了当作用群为Abel群时，一个驯服的度量极小动力系统（i）几乎是自守的（即它几乎是等度连续系统的一对一扩展），和（ii）承认一个唯一不变概率测度，使得相应的测度保持系统是测度-理论上同构于最大等度连续因子上的Haar测度系统。
Abstract A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.