Maximal pronilfactors and a topological Wiener-Wintner theorem (Q2677812)

A dynamical system (either topological or measurable) is called a \(d\)-step pronilsystem if it is an inverse limit of nilsystems of degree at most \(d\). In the measurable setting these systems arise naturally as characteristic factors for non-conventional ergodic averages, and in the topological setting these arise in connection with the higher order regionally proximal relations. Here both perspectives are studied in the setting of strictly ergodic systems, where a class of systems called \(CF_{\mathrm{Nil}}(k\)) systems are introduced, and characterised in two different ways. The first is via theorems of Wiener-Wintner type (see [\textit{N. Wiener} and \textit{A. Wintner}, Am. J. Math. 63, 415--426 (1941; JFM 67.0389.02)]) and the second is related to the \(k\)-cube uniquely ergodic systems introduced by \textit{Y. Gutman} and \textit{Z. Lian} [J. Funct. Anal. 284, No. 4, Article ID 109779, 54 p. (2023; Zbl 07634000)]. The class of systems studied is large in the sense that a deep result of \textit{B. Weiss} [Bull. Am. Math. Soc., New Ser. 13, 143--146 (1985; Zbl 0615.28012)] shows that any ergodic measure-preserving transformation is measurably isomorphic to a \(CF_{\mathrm{Nil}}(k\)) system. For \(k=1\) the results here provide a new condition equivalent to the property that every measurable eigenfunction has a continuous version for strictly ergodic systems.