Modified shrinking target problem for Matrix Transformations of Tori
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916099109421056 |
|---|---|
| author | Yuan, Na Wang, ShuaiLing |
| author_facet | Yuan, Na Wang, ShuaiLing |
| contents | We calculate the Hausdorff dimension of the fractal set \begin{equation*}
\Big\{\mathtt{x}\in \mathbb{T}^d: \prod_{1\leq i\leq d}|T_{β_i}^n(x_i)-x_i| < ψ(n) \text{ for infinitely many } n\in \mathbb{N}\Big\}, \end{equation*} where the $T_{β_i}$ is the standard $β_i$-transformation with $β_i>1$, $ψ$ is a positive function on $\mathbb{N}$ and $|\cdot|$ is the usual metric on the torus $\mathbb{T}$.
Moreover,
we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative
recurrence properties for matrix transformations of tori. Let $T$ be a $d\times d$ non-singular matrix with real coefficients. Then, $T$ determines a self-map of the $d$-dimensional torus $\mathbb{T}^d:=\mathbb{R}^d / \mathbb{Z}^d$.
For any $1\leq i \leq d$, let $ψ_i$ be a positive function on $\mathbb{N}$ and $Ψ(n):=(ψ_1(n),\dots, ψ_d(n))$ with $n\in \mathbb{N}$. We obtain the Hausdorff dimension of the fractal set
\begin{equation*}
\big\{\mathtt{x}\in \mathbb{T}^d: T^n(x)\in L(f_n(\mathtt{x}), Ψ(n)) \text{ for infinitely many } n\in \mathbb{N}\big\}, \end{equation*} where $L(f_n(\mathtt{x}, Ψ(n)))$ is a hyperrectangle and $\{f_n\}_{n\geq 1}$ is a sequence of Lipschitz vector-valued functions on $\mathbb{T}^d$ with a uniform Lipschitz constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07532 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modified shrinking target problem for Matrix Transformations of Tori Yuan, Na Wang, ShuaiLing Dynamical Systems 11K55, 37C45, 28A80 We calculate the Hausdorff dimension of the fractal set \begin{equation*} \Big\{\mathtt{x}\in \mathbb{T}^d: \prod_{1\leq i\leq d}|T_{β_i}^n(x_i)-x_i| < ψ(n) \text{ for infinitely many } n\in \mathbb{N}\Big\}, \end{equation*} where the $T_{β_i}$ is the standard $β_i$-transformation with $β_i>1$, $ψ$ is a positive function on $\mathbb{N}$ and $|\cdot|$ is the usual metric on the torus $\mathbb{T}$. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let $T$ be a $d\times d$ non-singular matrix with real coefficients. Then, $T$ determines a self-map of the $d$-dimensional torus $\mathbb{T}^d:=\mathbb{R}^d / \mathbb{Z}^d$. For any $1\leq i \leq d$, let $ψ_i$ be a positive function on $\mathbb{N}$ and $Ψ(n):=(ψ_1(n),\dots, ψ_d(n))$ with $n\in \mathbb{N}$. We obtain the Hausdorff dimension of the fractal set \begin{equation*} \big\{\mathtt{x}\in \mathbb{T}^d: T^n(x)\in L(f_n(\mathtt{x}), Ψ(n)) \text{ for infinitely many } n\in \mathbb{N}\big\}, \end{equation*} where $L(f_n(\mathtt{x}, Ψ(n)))$ is a hyperrectangle and $\{f_n\}_{n\geq 1}$ is a sequence of Lipschitz vector-valued functions on $\mathbb{T}^d$ with a uniform Lipschitz constant. |
| title | Modified shrinking target problem for Matrix Transformations of Tori |
| topic | Dynamical Systems 11K55, 37C45, 28A80 |
| url | https://arxiv.org/abs/2304.07532 |