On shrinking targets for linear expanding and hyperbolic toral endomorphisms
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916235390746624 |
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| author | Hu, Zhang-nan Persson, Tomas Wu, Wanlou Zhang, Yiwei |
| author_facet | Hu, Zhang-nan Persson, Tomas Wu, Wanlou Zhang, Yiwei |
| contents | Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $τ>0$, and a sequence $\{z_n\}$ of points in $\mathbb{T}^d$, let $W_τ$ be the set of points $x\in\mathbb{T}^d$ such that $T^n(x)\in B(z_n,e^{-nτ})$ for infinitely many $n\in\mathbb{N}$. The Hausdorff dimension of $W_τ$ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of $W_τ$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_τ$ only when $A$ is a $2 \times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02582 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On shrinking targets for linear expanding and hyperbolic toral endomorphisms Hu, Zhang-nan Persson, Tomas Wu, Wanlou Zhang, Yiwei Dynamical Systems 37C45, 37E05 Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $τ>0$, and a sequence $\{z_n\}$ of points in $\mathbb{T}^d$, let $W_τ$ be the set of points $x\in\mathbb{T}^d$ such that $T^n(x)\in B(z_n,e^{-nτ})$ for infinitely many $n\in\mathbb{N}$. The Hausdorff dimension of $W_τ$ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of $W_τ$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_τ$ only when $A$ is a $2 \times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$. |
| title | On shrinking targets for linear expanding and hyperbolic toral endomorphisms |
| topic | Dynamical Systems 37C45, 37E05 |
| url | https://arxiv.org/abs/2405.02582 |