Let $T:[0,1]^d \rightarrow[0,1]^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\mu$. Let $\left\{ \mathbf{ r}_n\right\}$ be a sequence of vectors satisfying the conditons that $\mathbf{ r}_n=\left(r_{ n, 1}, \ldots, r_{ n, d}\right) \in\left(\mathbb{ R}_{ \geq 0}\right)^d$, the sequence $\left\{ \frac{ \max _{ 1 \leq i \leq d}\hspace{ 1ex}r_{ n, i}}{ \min _{ 1 \leq i \leq d}\hspace{ 1ex}r_{ n, i}}\right\}$ is bounded and $\lim _{ n \rightarrow \infty} \max _{ 1 \leq i \leq d}r_{ n, i}=0$. Let $\left\{ \delta_n\right\}$ be a sequence of non-negative real numbers with $\lim _{ n \rightarrow \infty} \delta_n=0$. Under the assumptions that $\mu$ is exponentially mixing and its density is sufficiently regular, we prove that the $\mu$-measure of the following sets $$\mathcal{ R}^f\left(\left\{ \mathbf{ r}_n\right\}\right)=\left\{ \mathbf{ x} \in[0,1]^d: T^n \mathbf{ x} \in R\left(f(\mathbf{ x}), \mathbf{ r}_n\right) \text { for infinitely many } n \in \mathbb{ N} \right\} $$ and $$\mathcal{ R}^{ f \times}\left(\left\{ \delta_n\right\}\right)=\left\{ \mathbf{ x} \in[0,1]^d: T^n \mathbf{ x} \in H\left(f(\mathbf{ x}), \delta_n\right) \text { for infinitely many } n \in \mathbb{ N} \right\}$$ obeys zero-full laws determined by the convergence or divergence of natural volume sums. Here, $R(f(\mathbf{ x}), \mathbf{ r}_n)$ and $H(f(\mathbf{ x}), \delta_n)$ represent targets as, respectively, coordinate-parallel hyperrectangles with bounded aspect ratio, and hyperboloids, both centered at $f(\mathbf{ x})$. $f: [0,1]^d \rightarrow [0,1]^d$ is a piecewise Lipschitz vector function. Our results not only unify quantitative recurrence properties and the shrinking target problem for piecewise expanding maps on $[0,1]^d$, but also reveal that the two problems and cross-component recurrence can coexist in distinct directions on $[0,1]^d$.
Categories: math.DS