Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916402363891712 |
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| author | Kosloff, Zemer Sanadhya, Shrey |
| author_facet | Kosloff, Zemer Sanadhya, Shrey |
| contents | We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem.
We use the $2$-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages Kosloff, Zemer Sanadhya, Shrey Dynamical Systems Probability 28D05, 37A05, 37A50, 37A30, 60F05, 60G10 We show that for every ergodic and aperiodic probability preserving system $(X,\mathcal{B},m,T)$, there exists $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem. We use the $2$-dimensional result to resolve a question of Huang, Shao and Ye and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. Our methods also give the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. |
| title | Multidimensional local limit theorem in deterministic systems and an application to non-convergence of polynomial multiple averages |
| topic | Dynamical Systems Probability 28D05, 37A05, 37A50, 37A30, 60F05, 60G10 |
| url | https://arxiv.org/abs/2409.05087 |