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Modified shrinking target problem for Matrix Transformations of Tori

Na Yuan, ShuaiLing Wang

Abstract

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.

Modified shrinking target problem for Matrix Transformations of Tori

Abstract

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 is the standard -transformation with , is a positive function on and is the usual metric on the torus . 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 be a non-singular matrix with real coefficients. Then, determines a self-map of the -dimensional torus . For any , let be a positive function on and with . 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 is a hyperrectangle and is a sequence of Lipschitz vector-valued functions on with a uniform Lipschitz constant.
Paper Structure (8 sections, 13 theorems, 94 equations)

This paper contains 8 sections, 13 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Theorem \ref{['hypercube']}
  4. Proof of Theorem \ref{['main theorem1']} and Corollary \ref{['main_corollary']}
  5. The upper bound part
  6. The lower bound part
  7. The proof of Corollary \ref{['main_corollary']}
  8. Proof of Theorem \ref{['diagonalizable']}

Key Result

Theorem 1.1

Let $T$ be a real, non-singular matrix transformation of the torus $\mathbb{T}^d$. Suppose that $T$ is diagonal and all eigenvalues $\beta_1$, $\beta_2$, $\dots$, $\beta_d$ are of modulus strictly larger than $1$. Assume that $1<\beta_1\leq \beta_2\leq \dots\leq \beta_d$. Then

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1: Rnyi1957RepresentationsFR
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 10 more