Non-commuting transformations with non-converging 2-fold ergodic averages
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910551577198592 |
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| author | Ryzhikov, Valery V. |
| author_facet | Ryzhikov, Valery V. |
| contents | In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with $|N_{m+1}|/|N_m|\ \to\infty$ such that $T^iA\cap S^iA=ϕ$ for $i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. For the corresponding deterministic Gaussian and Poisson suspensions $S_\circ,T_\circ$ over $S,T$ for some $f\in L_\infty$ there is no limit of $ \frac 1 N \sum_{n=1}^N T_\circ^nf\,S_\circ^nf$ in $ L_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13741 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-commuting transformations with non-converging 2-fold ergodic averages Ryzhikov, Valery V. Dynamical Systems In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with $|N_{m+1}|/|N_m|\ \to\infty$ such that $T^iA\cap S^iA=ϕ$ for $i\in [N_{4k}, N_{4k+1}]$ and $T^iA=S^iA$ for $i\in [N_{4k+2}, N_{2k+3}]$. For the corresponding deterministic Gaussian and Poisson suspensions $S_\circ,T_\circ$ over $S,T$ for some $f\in L_\infty$ there is no limit of $ \frac 1 N \sum_{n=1}^N T_\circ^nf\,S_\circ^nf$ in $ L_2$. |
| title | Non-commuting transformations with non-converging 2-fold ergodic averages |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2407.13741 |