Quantitative twisted recurrence properties for piecewise expanding maps on $[0,1]^d$

Abstract

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 $[0,1]^d$.