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

TL;DR

This work provides a unified, quantitative treatment of twisted recurrence and shrinking-target phenomena for piecewise expanding maps on $[0,1]^d$, establishing zero-full laws for two families of targets: coordinate-parallel hyperrectangles and hyperboloids. Under exponential mixing and regular density assumptions on the invariant measure, the authors prove that the measure of the twisted recurrence sets $\mathcal{R}^f(\{\mathbf{r}_n\})$ and $\mathcal{R}^{f\times}(\{\delta_n\})$ is determined by the convergence or divergence of natural volume sums, with refined results when $f$ is coordinate-wise or when stronger density hypotheses hold. The proofs hinge on local approximations of target sets, Zygmund differentiation for density control, mixing properties, and second-moment methods (Paley-Zygmund) to obtain full-measure results, while accommodating cross-component recurrence in distinct directions. The results fuse recurrence, shrinking-target, and cross-component recurrence into a cohesive framework, offering new insight into how these phenomena interact in high-dimensional non-conformal settings and across different target geometries.

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 $μ$. 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 $\left\{δ_n\right\}$ be a sequence of non-negative real numbers with $\lim _{n \rightarrow \infty} δ_n=0$. Under the assumptions that $μ$ is exponentially mixing and its density is sufficiently regular, we prove that the $μ$-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 $$\mathcal{R}^{f \times}\left(\left\{δ_n\right\}\right)=\left\{\mathbf{x} \in[0,1]^d: T^n \mathbf{x} \in H\left(f(\mathbf{x}), δ_n\right) \text { for infinitely many } n \in \mathbb{N} \right\}$$ obeys zero-full laws determined by the convergence or divergence of natural volume sums. Here, $R(f(\mathbf{x}), \mathbf{r}_n)$ and $H(f(\mathbf{x}), δ_n)$ 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 $[0,1]^d$.

Figures (1)

• Figure 1: The finite family $\left(U_i\right)_{i=1}^{10}$ of non-empty, pairwise disjoint, and connected open subsets in $[0,1)^2$ for $T \mathbf{x}=(\frac{3}{2}x_1+x_2 \bmod 1,\sqrt{2}x_1-2x_2 \bmod 1)$

Theorems & Definitions (66)

• Definition 1.1: 7
• Remark 1
• Definition 1.2: Cylinders of order $n$
• Proposition 1.1: 16
• Definition 1.3: $\phi(n)$-mixing
• Definition 1.4: Piecewise Lipschitz vector function
• Definition 1.5: $\mu \circ f^{-1} \ll \mu$
• Remark 2
• Remark 3
• Theorem 1.2