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

Abstract

Let $T:[0,1{]}^d\to [0,1{]}^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\hat{I}¼$. Let $\left\{{\mathbf{r}}_n\right\}$ be a sequence of vectors satisfying the conditons that ${\mathbf{r}}_n=\left(r_{n,1},\dots ,r_{n,d}\right)\in {\left({\mathbb{R}}_{\ge 0}\right)}^d$, the sequence $\left\{\frac{{\max }_{1\le i\le d}{r}_{n,i}}{{\min }_{1\le i\le d}{r}_{n,i}}\right\}$ is bounded and ${\lim }_{n\to \infty }{\max }_{1\le i\le d}{r}_{n,i}=0$. Let $\left\{\hat{I}{´}_n\right\}$ be a sequence of non-negative real numbers with ${\lim }_{n\to \infty }\hat{I}{´}_n=0$. Under the assumptions that $\hat{I}¼$ is exponentially mixing and its density is sufficiently regular, we prove that the $\hat{I}¼$-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 $${\mathcal{R}}^{f\times }\left(\left\{\hat{I}{´}_n\right\}\right)=\left\{\mathbf{x}\in [0,1{]}^d:T^n\mathbf{x}\in H\left(f(\mathbf{x}),\hat{I}{´}_n\right)\text{ for infinitely many }n\in \mathbb{N}\right\}$$ obeys zero-full laws determined by the convergence or divergence of natural volume sums. Here, $R(f(\mathbf{x}),{\mathbf{r}}_n)$ and $H(f(\mathbf{x}),\hat{I}{´}_n)$ represent targets as, respectively, coordinate-parallel hyperrectangles with bounded aspect ratio, and hyperboloids, both centered at $f(\mathbf{x})$. $f:[0,1{]}^d\to [0,1{]}^d$ 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 $[0,1{]}^d$.

37pages, 1 figure. arXiv admin note: text overlap with arXiv:2302.05149, arXiv:2208.06112 by other authors