Topological Wiener-Wintner ergodic theorem with polynomial weights

TL;DR

This paper proves a topological Wiener-Wintner ergodic theorem with polynomial weights for certain uniquely ergodic systems, enabling the construction of fully oscillating sequences on nilmanifolds.

Contribution

It establishes a new ergodic theorem with polynomial weights under spectral conditions, extending the understanding of oscillating sequences in dynamical systems.

Findings

The theorem applies to ergodic nilsystems.

Constructs fully oscillating sequences on nilmanifolds.

Provides conditions for polynomial-weighted ergodic averages.

Abstract

For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.