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