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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2311.01031 |
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| _version_ | 1866917801529180160 |
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| author | He, Yubin |
| author_facet | He, Yubin |
| contents | For $ β>1 $ let $ T_β$ be the $β$-transformation on $ [0,1) $. Let $ β_1,\dots,β_d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define
\[W(\mathcal P)=\{\textbf{x}\in[0,1)^d:(T_{β_1}\times\cdots \times T_{β_2})^n(\textbf{x})\in P_n\text{ infinitely often}\}.\]
When each $ P_n $ is a hyperrectangle with sides parallel to the axes, the 'rectangle to rectangle' mass transference principle by Wang and Wu [Math. Ann. 381 (2021)] is usually employed to derive the lower bound for $\mathrm{dim_H} W(\mathcal P)$, where $\mathrm{dim_H}$ denotes the Hausdorff dimension. However, in the case where $ P_n $ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining $\mathrm{dim_H} W(\mathcal P)$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{dim_H} W(\mathcal P)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01031 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shrinking parallelepiped targets in beta-dynamical systems He, Yubin Dynamical Systems For $ β>1 $ let $ T_β$ be the $β$-transformation on $ [0,1) $. Let $ β_1,\dots,β_d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define \[W(\mathcal P)=\{\textbf{x}\in[0,1)^d:(T_{β_1}\times\cdots \times T_{β_2})^n(\textbf{x})\in P_n\text{ infinitely often}\}.\] When each $ P_n $ is a hyperrectangle with sides parallel to the axes, the 'rectangle to rectangle' mass transference principle by Wang and Wu [Math. Ann. 381 (2021)] is usually employed to derive the lower bound for $\mathrm{dim_H} W(\mathcal P)$, where $\mathrm{dim_H}$ denotes the Hausdorff dimension. However, in the case where $ P_n $ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining $\mathrm{dim_H} W(\mathcal P)$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{dim_H} W(\mathcal P)$. |
| title | Shrinking parallelepiped targets in beta-dynamical systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2311.01031 |