Infinite-step nilsystems, independence and complexity

By P. Dong, Sebastian Donoso, Alejandro Maass, S. Shao, Xiangdong Ye. Ergodic Theory and Dynamical Systems, 2011

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ade41746f0d5edb896e3023ff81f7590ce28800