Complexity of Nilsystems and systems lacking nilfactors

TL;DR

This paper investigates the complexity and limitations of nilsystems in ergodic and topological dynamics, revealing that many systems lack higher order nilfactors and exhibit specific spectral and polynomial growth obstructions.

Contribution

It demonstrates that many natural dynamical systems contain only rotations as nilfactors and identifies spectral and polynomial growth obstructions to higher order nilsystems.

Findings

Many systems contain only rotations as nilfactors.

Spectral obstructions prevent higher order nilsystems from being factors.

Nilsystems exhibit polynomial complexity with explicit degree bounds.

Abstract

Nilsystems are a natural generalization of rotations and arise in various contexts, including in the study of multiple ergodic averages in ergodic theory, in the structural analysis of topological dynamical systems, and in asymptotics for patterns in certain subsets of the integers. We show, however, that many natural classes in both measure preserving systems and topological dynamical systems contain no higher order nilsystems as factors, meaning that the only nilsystems they contain as factors are rotations. In the ergodic setting, we show that there are spectral obstructions that give rise to this behavior. In the topological setting, nilsystems have a particular type of complexity of polynomial growth, where the polynomial (with explicit degree) is an asymptotic both from below and above. We also deduce several ergodic and topological applications of these results.