Maximal pronilfactors and a topological Wiener—Wintner theorem

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.