For strictly ergodic systems, we introduce the class of continuous to $k$-nil 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. We characterize a continuous to $k$-nil system in terms of its $(k+1)$-dynamical cubespace. In particular, for $k=1$, this gives a new condition equivalent to the property that every measurable eigenfunction has a continuous version. We also show that the continuous to $k$-nil systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. Joint work with Zhengxing Lian.
Meeting ID: 838 4895 5300 Passcode: 797123