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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/2407.19444 |
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| _version_ | 1866917736141029376 |
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| author | Fish, Alexander Skinner, Sean |
| author_facet | Fish, Alexander Skinner, Sean |
| contents | We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,μ,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ μ\left( \bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left( \bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] for all $k,n_1,\dots,n_k \in \mathbb{N}$. As a corollary we show that a set $R\subset \mathbb{N}$ is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and it's corollary partially answer two questions of Moreira. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An inverse of Furstenberg's correspondence principle and applications to nice recurrence Fish, Alexander Skinner, Sean Dynamical Systems We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,μ,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ μ\left( \bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left( \bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] for all $k,n_1,\dots,n_k \in \mathbb{N}$. As a corollary we show that a set $R\subset \mathbb{N}$ is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and it's corollary partially answer two questions of Moreira. |
| title | An inverse of Furstenberg's correspondence principle and applications to nice recurrence |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2407.19444 |