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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.10728 |
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| _version_ | 1866913431413587968 |
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| author | Huang, Wen Shao, Song Ye, Xiangdong |
| author_facet | Huang, Wen Shao, Song Ye, Xiangdong |
| contents | It is shown that there exist a probability space $(X,{\mathcal X},μ)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},μ)$ with $h_μ(X,T)=h_μ(X,S)=0$, and $f, g \in L^\infty(X,μ)$ such that the limit \begin{equation*}
\lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \end{equation*} does not exist in $L^2(X,μ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10728 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample on multiple convergence without commutativity Huang, Wen Shao, Song Ye, Xiangdong Dynamical Systems It is shown that there exist a probability space $(X,{\mathcal X},μ)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},μ)$ with $h_μ(X,T)=h_μ(X,S)=0$, and $f, g \in L^\infty(X,μ)$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \end{equation*} does not exist in $L^2(X,μ)$. |
| title | A counterexample on multiple convergence without commutativity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2407.10728 |