Infinite-step nilsystems, independence and complexity
Infinite-step nilsystems, independence and complexity
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DOI：
10.1017/s0143385711000861
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发表时间：
2011-05
影响因子：
0.9
通讯作者：
P. Dong;S. Donoso;A. Maass;S. Shao;X. Ye
中科院分区：
文献类型：
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作者：
P. Dong;S. Donoso;A. Maass;S. Shao;X. Ye
∞阶零系统是极小零系统的逆极限。本文证明了一个极小远端系统是∞阶零系统，当且仅当它不存在具有任意长有限ip无关集的非平凡对。此外，证明了任意具有任意长有限ip无关集的无非平凡对的最小系统是其最大∞-阶零因子的几乎一一扩展，并且证明了每个不变遍历测度与某个∞-阶零系统上的Haar测度在可测意义上同构。这样一个系统是否具有独特的遍历性，这个问题仍然悬而未决。此外，计算了∞阶零系统的拓扑复杂度，证明了它对每个非平凡开覆盖都是多项式。
Abstract An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal ∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some ∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an ∞-step nilsystem is computed, showing that it is polynomial for each non-trivial open cover.