Hausdorff dimension of recurrence sets for matrix transformations of tori
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916117663973376 |
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| author | Hu, Zhangnan Li, Bing |
| author_facet | Hu, Zhangnan Li, Bing |
| contents | Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|λ_1|\le|λ_2|\le\dots\le|λ_d|$. We investigate the Hausdorff dimension of the recurrence set
\[R(ψ):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,ψ(n)) {\rm ~for~infinitely~ many~}n\}\]
for $α\ge\log|λ_d/λ_1|$, where $ψ$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $α=\liminf\limits_{n\to\infty}\frac{-\log ψ(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04810 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of recurrence sets for matrix transformations of tori Hu, Zhangnan Li, Bing Dynamical Systems 37C45, 37B20, 28A80 Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|λ_1|\le|λ_2|\le\dots\le|λ_d|$. We investigate the Hausdorff dimension of the recurrence set \[R(ψ):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,ψ(n)) {\rm ~for~infinitely~ many~}n\}\] for $α\ge\log|λ_d/λ_1|$, where $ψ$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $α=\liminf\limits_{n\to\infty}\frac{-\log ψ(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula. |
| title | Hausdorff dimension of recurrence sets for matrix transformations of tori |
| topic | Dynamical Systems 37C45, 37B20, 28A80 |
| url | https://arxiv.org/abs/2402.04810 |