Infinite-step nilsystems, independence and complexity

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.