Modified shrinking target problem for Matrix Transformations of Tori

Let $(X,d)$ be a metric space, and let $T:X\to X$ be a transformation. If $\mu$ is a $T$-invariant Borel probability measure on $X$ and $T$ is ergodic with respect to the measure $\mu$, Birkhoff’s ergodic theorem implies that for any ball $B\subset X$ of positive measure, the subset $S=\{x\in X:T^{n}x\in B\text{\ for \ i.m.\ }n\in\mathbb{N}\}$ has full $\mu$-measure. Here and throughout the paper, ‘i.m.’ stands for ‘infinitely many’. This means that the trajectories of almost all points will enter the ball $B$ infinitely often. In general, one may wonder what will happen if $B$ shrinks with time. Let $\psi$ be a function on $\mathbb{N}$ and $z\in X$, the investigation of the size in terms of measure and dimension of the set

is called shrinking target problems by Hill and Velani [13], where $B(z,\psi(n))\subset X$ is a ball of center $z$ and radius $\psi(n)$. On the other hand, motivated by Poincaré’s recurrence theorem in dynamical systems, one is also interested in the quantitative recurrence set(see[5, 6])

The measures and dimensions of the fractal sets $S(\psi,z)$ and $R(\psi)$ have already been extensively investigated in many dynamical systems. For the measure aspect, the reader is referred to [1, 2, 9, 10, 12, 16, 17, 19, 20, 21] and references therein. With regards to the dimension aspect, let us cite some but far from all concrete cases. The dimension aspect has been studied for expanding rational maps of Julia sets [13, 14], matrix transformations of tori [15, 21], $\beta$-transformations [18, 27, 29], conformal iterated function systems [24, 26, 22]. For the more results, the reader is referred to [23, 3, 4, 33] and references therein.

It should be observed that in many systems, the fractal dimensional formulae for the two sets $S(\psi)$ and $R(\psi)$ are the same. In [28, 31, 32, 34], the dimensions of $S(\psi)$ and $R(\psi)$ were unified in some dynamical systems by using a Lipschitz function. We are interested in finding out if they can be unified when $T$ is a matrix transformation on the torus $\mathbb{T}^{d}$.

We start by laying out some necessary definitions and notations. Let $\mathbb{T}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d}$ be a $d$-dimensional torus endowed with the usual metric in $\mathbb{T}^{d}$ induced by the Euclidean metric. Let $T$ be a $d\times d$ non-singular matrix with real coefficients. Then, $T$ determines a self-map of the $d$-dimensional torus $\mathbb{T}^{d}$, namely, for any $\mathtt{x}\in\mathbb{T}^{d}$, $T:\mathtt{x}\to T(\mathtt{x})\ (\bmod 1)$. In what follows, $T$ will denote both the matrix and the transformation. Denote by $T^{n}$ the $n$-th iteration of the transformation $T$ with $n\in\mathbb{N}$. Let $\mathbb{R}^{+}$ be the set of positive numbers.

We explore two special sets that are relevant to the classical theory of Diophantine approximation. For any $1\leq i\leq d$, let $\psi_{i}$ be a positive function on $\mathbb{N}$ and $\Psi(n):=(\psi_{1}(n),\dots,\psi_{d}(n))$ with $n\in\mathbb{N}$. For any $\mathtt{y}=(y_{1},y_{2},\dots,y_{d})\in\mathbb{T}^{d}$, let

and

where $|\cdot|$ is the usual metric on $\mathbb{T}$. Then $L(\mathtt{y},\psi(n))$ is a hyperrectangle and $P(\mathtt{y},\psi(n))$ is a hyperboloid. Define

The set $H_{d}(T,\psi)$ is intimately related to sets studied within the multiplicative theory of Diophantine approximation.

Let $\{f_{n}\}_{n\geq 1}$ be a sequence of vector-valued functions on $\mathbb{T}^{d}$ with a uniform Lipsitz constant, i.e., there exists $c\in\mathbb{R}^{+}$ such that for any $n\in\mathbb{N}$ and $\mathtt{x},\mathtt{y}\in\mathbb{T}^{d}$,

where we also use $|\cdot|$ for the usual metric on $\mathbb{T}^{d}$ induced by the Euclidean metric. Then, (1.1) means that for any $n\in\mathbb{N}$ and $x,y\in\mathbb{T}$,

where $f^{(i)}_{n}$ is the function $f_{n}$ restricted to the direction of the $i$-th axis.

Define the modified shrinking target set associated to $\Psi$ and $\{f_{n}\}_{n\geq 1}$ by

If $\psi_{1}=\psi_{2}=\dots=\psi_{d}=\psi$, we write

Note that if $f_{n}=y$ for all $n\in\mathbb{N}$, where $\mathtt{y}$ is a fixed point in $\mathbb{T}^{d}$, then the fractal set $W(T,\psi,\{f_{n}\})$ is an analogue of the sets explored by the classical simultaneous theory of Diophantine approximation.

Denote by $\dim_{\mathrm{H}}$ the Hausdorff dimension. Let

Our results are listed below.

Let $\mathcal{C}(\Psi)$ be the set of accumulation points $\mathbf{t}=(t_{1},\dots,t_{d})$ of the sequence

The following Theorem 1.2 gives the value of $\dim_{\mathrm{H}}W(T,\Psi,\{f_{n}\})$ when $T$ is a real, non-singular diagonal matrix transformation of the torus $\mathbb{T}^{d}$.

If for each $1\leq i,j\leq d$, $\psi_{i}=\psi_{j}=\psi$ in Theorem 1.2, we can obtain the following Corollary 1.4.

The following Theorem 1.6 extends Corollary 1.4 from diagonal matrix to the matrix which can be diagonalizable over $\mathbb{Z}$.

Our paper is organized as follows. Section 2 begins with some preparations on $\beta$-transformation and some useful Lemma. Section 3 provides the proof of Theorem 1.1. The proof of Theorem 1.2 and Corollary 1.4 can be found in Section 4. More precisely, we establish the upper bound of $\dim_{\mathrm{H}}W(T,\Psi,\{f_{n}\})$ in Section 4.1, and obtain the lower bound of $\dim_{\mathrm{H}}W(T,\Psi,\{f_{n}\})$ in Section 4.2. Section 4.3 gives the proof of Corollary 1.4. The proof of Theorem 1.6 is presented in Sections 5.

Let $\beta>1$. Define the $\beta$-transformation $T_{\beta}:[0,1)\rightarrow[0,1)$ by

where $\lfloor\zeta\rfloor$ stands for the largest integer no more than $\zeta$. The system $([0,1),T_{\beta})$ is called $\beta$-dynamical system. Then every real number $x\in[0,1)$ can be expressed uniquely as a finite or an infinite series

where, for $n\geq 1$, $\epsilon_{n}(x,\beta)=\lfloor\beta T_{\beta}^{n-1}x\rfloor$ is called $n$-th digit of $x$ (with respect to base $\beta$). Then formula (2.1) or the digit sequence $\varepsilon(x,\beta):=(\epsilon_{1}(x,\beta),\epsilon_{2}(x,\beta),\dots)$ is called the $\beta$-expansion of $x$. Sometimes we rewrite (2.1) as

It is clear that, for $n\geq 1$, the $n$-th digit $\epsilon_{n}(x,\beta)\in\mathcal{A}_{\beta}=\{0,1,\dots,\lceil\beta-1\rceil\}$, where $\lceil\beta-1\rceil=\min\{j\in\mathbb{N}:j\geq\beta-1\}$. While, not all sequence $\omega\in\mathcal{A}^{\mathbb{N}}_{\beta}$ would be a $\beta$-expansion of some $x\in[0,1]$. We call a finite or an infinite sequence $(\epsilon_{1},\epsilon_{2},\dots)$ admissible if there exists an $x\in[0,1)$ such that the $\beta$-expansion of $x$ begins with $(\epsilon_{1},\epsilon_{2},\dots)$. For $n\geq 1$, let $\Sigma_{\beta}^{n}$ be the set of all admissible sequences of length $n$.

The following widely recognized result from Rényi.

The following property of full cylinders plays an important role in the proofs of Theorem 1.2.

We can get the following lemma by similar idea as in the proof of [32, Lemma 4].

The following lemma shows that the Hausdorff dimension of a set is invariant under bi-Lipschitz maps.

Now we will introduce the Mass Transference Principle for rectangles that is useful to get the lower bound of $\dim_{\mathrm{H}}W(T,\Psi,\{f_{n}\})$ in Theorem 1.2. For our purpose, we only need the following special case of [30, Theorem 3.4].

We begin by presenting some lemma that we will use to prove Theorem 1.1.

Let $\delta>0$ and define

The following result is important in the proofs of Theorem 1.1.

For a cube $B\subset\mathbb{T}^{d}$, write it as

Define

By Lemma 3.2, we obtain a cover of $E_{n}(\delta)$ in the following Lemma 3.3.

We now begin to prove Theorem 1.1. Recall that

Let

Then $H_{d}(T,\psi)=\limsup_{n\to\infty}E_{n}(\psi)$.

We divided the proof into two parts: the upper bound and the lower bound, as is typical when calculating the Hausdorff dimension of a set.

As usual, the upper bound is obtained by finding an efficient cover of $H_{d}(T,\psi)$. First, for sufficiently large $n$, we find an efficient cover of $E_{n}(\psi)$. By Lemmas 3.2 and 3.3, with $\delta=\psi(n)$ and $n>\log_{\beta_{1}}2$, for any $s\in(d-1,d)$, there exists a collection $\mathcal{B}$ of cubes $B$ such that

and

By Lemma 3.1, we obtain that $E_{n,\beta_{i}}(\mathtt{w}_{i},a_{B},b_{B})$ can be covered by $4$ intervals of length $(b_{B}-a_{B})\beta_{i}^{-n}$ for any $1\leq i\leq d$. Thus, $\prod_{i=1}^{d}E_{n,\beta_{i}}(\mathtt{w}_{i},a_{B},b_{B})$ can be covered by $4^{d}$ hyperrectangles, and the side length of these hyperrectangles in the direction of the $i$-th axis is $(b_{B}-a_{B})\beta^{-n}_{i}=|B|\beta_{i}^{-n}$.

We now cover $\prod_{i=1}^{d}E_{n,\beta_{i}}(\mathtt{w}_{i},a_{B},b_{B})$ by balls with diameter equal to $\beta_{d}^{-n}$. Note that for any $1\leq i\leq d$, $\beta_{i}^{-n}\geq\beta_{d}^{-n}$, then we can find a collection $\mathcal{C}_{n}$ of balls with diameter $\beta_{d}^{-n}|B|$ that cover $\prod_{i=1}^{d}E_{n,\beta_{i}}(\mathtt{w}_{i},a_{B},b_{B})$ with

Hence for any $N>\log_{\beta_{1}}2$,

Given $\varepsilon>0$, we choose $N$ large enough so that for any $n\geq\max\{N,\log_{\beta_{1}}2\}$ and any ball $B\in\mathcal{B}$, $\beta_{d}^{-n}|B|<\varepsilon$. Then, according to the definition of the $s$-dimensional Hausdorff measure, for any $s>0$

On the one hand, by Lemma 2.1, $\#\Sigma^{n}_{\beta}\leq\frac{\beta^{n+1}}{\beta-1}$. This together with (3.3) and (3) implies there exists $c\in\mathbb{R}$ such that for any $s\in(d-1,d)$ and sufficiently large $N$

Therefore, for any

$\mathcal{H}^{s}(H_{d}(T,\psi))=0$, which implies that

Now, we will give the proof of the lower bound of $\dim_{\mathrm{H}}H_{d}(T,\psi)$.

For any $n\geq 1$, letting $f_{n}(x)=x$ and $d=1$ in Theorem 1.2, we have that

where $\tau=\liminf_{n\to\infty}\frac{-\log\psi(n)}{n}$.

For any $x_{d}\in H_{1}(T,\psi)$,

which follows that