Metastability in the Furstenberg-Zimmer Tower
Metastability in the Furstenberg-Zimmer Tower
复制标题
Furstenberg-Zimmer 塔的亚稳态
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
H. Towsner
中科院分区：
文献类型：
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作者：
J. Avigad;H. Towsner
根据Furstenberg-Zimmer结构定理，每个保测度系统都有一个极大远因子，并且相对于该因子是弱混合的。Furstenberg和Katznelson使用这种保测度系统的结构分析为Szemeredi定理提供了一个清晰的证明。Beleznay和Foreman证明了，一般来说，可分保测系统的极大远因子的超限构造可以延伸到可数序数中任意远。在这里，我们表明，Furstenberg-Katznelson证明不需要最大远端因子的全部强度，在这个意义上，证明只依赖于其属性的组合弱化。我们表明，这种组合较弱的性质得到相当低的超限建设，即，由ω th水平。
According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemeredi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the ωω ω th level.