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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.17267 |
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Table of Contents:
- A topological dynamical system $(X,T)$ is called CF-Nil($k$) if it is strictly ergodic and the maximal measurable and maximal topological $k$-step pro-nilfactors coincide as measure preserving systems. Through constructing specific ``CF-Nil'' models, we prove that for any ergodic system $(X,\mathcal{X},μ,T)$, any nilsequence $\{ψ(m,n)\}_{m,n\in\mathbb{Z}}$ and any $f_1,\dots,f_d\in L^{\infty}(μ)$, the averages \begin{equation*} \dfrac{1}{N^{2}} \sum_{m,n=0}^{N-1} ψ(m,n)\prod_{j=1}^{d}f_{j}(T^{m+jn}x) \end{equation*} converge pointwise as $N$ goes to infinity. Moreover, we show the $L^2$-convergence of a certain two-dimensional averages for non-commuting transformations without zero entropy condition.