On recurrence sets for toral endomorphisms
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912196474175488 |
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| author | Hu, Zhangnan Persson, Tomas |
| author_facet | Hu, Zhangnan Persson, Tomas |
| contents | Let $A$ be a $2\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, induced by $A$. Given $τ>0$, let
\[
R_τ=\{\, x\in \mathbb{T}^2 : T^nx\in B(x,e^{-nτ})~\mathrm{infinitely ~many}~n\in\mathbb{N} \,\}.
\]
We calculated the Hausdorff dimension of $R_τ$, and also prove that $R_τ$ has a large intersection property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11476 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On recurrence sets for toral endomorphisms Hu, Zhangnan Persson, Tomas Dynamical Systems 37C45, 37D20, 28A80 Let $A$ be a $2\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, induced by $A$. Given $τ>0$, let \[ R_τ=\{\, x\in \mathbb{T}^2 : T^nx\in B(x,e^{-nτ})~\mathrm{infinitely ~many}~n\in\mathbb{N} \,\}. \] We calculated the Hausdorff dimension of $R_τ$, and also prove that $R_τ$ has a large intersection property. |
| title | On recurrence sets for toral endomorphisms |
| topic | Dynamical Systems 37C45, 37D20, 28A80 |
| url | https://arxiv.org/abs/2501.11476 |