Infinite-step nilsystems, independence and complexity

TL;DR

This paper characterizes minimal distal systems as $ $-step nilsystems based on IP-independence sets, explores their ergodic measures, and computes their topological complexity, advancing understanding of their structure and dynamics.

Contribution

It establishes a characterization of $ $-step nilsystems via IP-independence, links systems without such sets to their nilfactors, and calculates their polynomial topological complexity.

Findings

Minimal distal systems are $ $-step nilsystems iff they lack nontrivial pairs with long IP-independence sets.

Such systems are almost one-to-one extensions of their maximal $ $-step nilfactors.

The topological complexity of $ $-step nilsystems is polynomial for any nontrivial open cover.

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.