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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2410.10211 |
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| _version_ | 1866929540949868544 |
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| author | He, Yubin |
| author_facet | He, Yubin |
| contents | Let $ ([0,1]^d,T,μ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $ μ$ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under mild conditions on the underlying dynamical system, we obtain a strong dynamical Borel-Cantelli lemma for recurrence: For any sequence $ \{R_n\} $ of hyperrectangles with sides parallel to the axes and centered at the origin,
\[\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}χ_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $ μ$-a.e.$\textbf{x}$},\]
where $ \textbf{x}\in[0,1]^d $ and $ R_k+\textbf{x} $ is the translation of $ R_k $. The result applies to Gauss map, $β$-transformation and expanding toral endomorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10211 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative recurrence properties and strong dynamical Borel-Cantelli lemma for dynamical systems with exponential decay of correlations He, Yubin Dynamical Systems Let $ ([0,1]^d,T,μ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $ μ$ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under mild conditions on the underlying dynamical system, we obtain a strong dynamical Borel-Cantelli lemma for recurrence: For any sequence $ \{R_n\} $ of hyperrectangles with sides parallel to the axes and centered at the origin, \[\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}χ_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $ μ$-a.e.$\textbf{x}$},\] where $ \textbf{x}\in[0,1]^d $ and $ R_k+\textbf{x} $ is the translation of $ R_k $. The result applies to Gauss map, $β$-transformation and expanding toral endomorphisms. |
| title | Quantitative recurrence properties and strong dynamical Borel-Cantelli lemma for dynamical systems with exponential decay of correlations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2410.10211 |