Maximal pronilfactors and a topological Wiener—Wintner theorem

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