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arXiv · 1105.3584

Infinite-step nilsystems, independence and complexity

P. D. Dong·S. Donoso·A. Maass·S. Shao·X. D. Ye
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Abstract

An $\infty$-step nilsystem is an inverse limit of minimal nilsystems. In this article is shown that a minimal distal system is an $\infty$-step nilsystem if and only if it has no nontrivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without nontrivial pairs with arbitrarily long finite IP-independence sets is an almost one to one extension of its maximal $\infty$-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some $\infty$-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an $\infty$-step nilsystem is computed, showing that it is polynomial for each nontrivial open cover.

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P. D. Dong, S. Donoso, A. Maass, S. Shao, X. D. Ye. 2011-05-18. Infinite-step nilsystems, independence and complexity. https://arxiv.org/abs/1105.3584

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