Maximal pronilfactors and a topological Wiener-Wintner theorem

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Gutman, Yonatan, Lian, Zhengxing
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.05.2022
Subjects
Eigenvectors
Ergodic processes
Theorems
Topology
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Summary: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.
ISSN:2331-8422