Let $T:[0,1]^d \rightarrow [0,1]^d$ be a piecewise expanding linear map such that each domain of linearity is a coordinate-parallel hyperrectangle that T sends to $[0,1]^d$. Let $\{\mathbf {r}_n\}$ be a sequence of vectors with $\mathbf {r}_n=(r_{n, 1}, \ldots , r_{n, d}) \in (\mathbb {R}_{\geq 0})^d$. Let $m_d$ denote d-dimensional Lebesgue measure. Let $\{f_n\}$ be a sequence of piecewise Lipschitz coordinate-wise vector functions on $[0,1]^d$ with a uniform Lipschitz constant. For any given $\epsilon> 0$, we obtain a unified counting function for the shrinking target and recurrence problems:

$$ \begin{align*}\sum_{n=1}^N \mathbb{1}_{R(f_n(\mathbf{x}), \mathbf{r}_n)}(T^n \mathbf{x})=\Psi(N)+O(\Psi^{1 / 2}(N)(\log \Psi(N))^{3 / 2+\varepsilon})\end{align*} $$

for $m_d$-almost every $\mathbf {x} \in [0,1]^d$ and for all $N\in \mathbb N$, where $\Psi (N):=2^d \sum _{n=1}^N r_{n, 1} \cdots r_{n, d}$ and $R(f_n(\mathbf {x}), \mathbf {r}_n)$ is a coordinate-parallel hyperrectangle centred at $f_n(\mathbf {x})$. Our result gives a unified statement to show that the quantitative dynamical Borel–Cantelli lemma can govern both the shrinking target and recurrence problems, thereby eliminating the need for separate treatments and different proofs. This result can also apply to the situation where both problems are present simultaneously along different directions in $[0,1]^d$.

For $ \beta>1 $, let $ T_\beta $ be the $\beta $-transformation on $ [0,1) $. Let $ \beta _1,\ldots ,\beta _d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define

$$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}}$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}} W(\mathcal P)$

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$, define the shrinking target set

$$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert <e^{-S_nf(x)}\\[1ex] \lvert T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \mathbb N \right\}, \end{align*} $$

where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.

In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\rightarrow \unicode[STIX]{x1D6FD}x\;\text{mod}\;1$ . Let $\unicode[STIX]{x1D6F9}_{i}$ ( $i=1,2$ ) be two positive functions on $\mathbb{N}$ such that $\unicode[STIX]{x1D6F9}_{i}\rightarrow 0$ when $n\rightarrow \infty$ . We determine the Lebesgue measure and Hausdorff dimension for the $\limsup$ set

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}_{1},\unicode[STIX]{x1D6F9}_{2})=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-x_{0}|<\unicode[STIX]{x1D6F9}_{1}(n),|T_{\unicode[STIX]{x1D6FD}}^{n}y-y_{0}|<\unicode[STIX]{x1D6F9}_{2}(n)\text{ for infinitely many }n\in \mathbb{N}\}\end{eqnarray}$$

$x_{0},y_{0}\in [0,1]$