Maximal pronilfactors and a topological Wiener-Wintner theorem

TL;DR

This paper introduces CF-Nil($k$) systems in strictly ergodic systems, showing their equivalence to systems where certain averages converge everywhere and characterizing them via dynamical cubespaces, with implications for eigenfunctions.

Contribution

It defines CF-Nil($k$) systems, proves their abundance, and characterizes them through dynamical cubespaces, linking measure-theoretic and topological properties.

Findings

CF-Nil($k$) systems are abundant in strictly ergodic systems.

They are exactly the systems where the $k$-step nilsequence Wiener-Wintner averages converge everywhere.

For $k=1$, every measurable eigenfunction has a continuous version.

Abstract

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil($k$) systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof we establish that pronilsystems are $coalescent$ both in the measurable and topological categories. In addition, we characterize a CF-Nil($k$) system in terms of its $(k+1)$-$th\ dynamical\ cubespace$. In particular, for $k=1$, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version.