Infinite-step nilsystems, independence and complexity

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.