Quantitative recurrence properties and strong dynamical Borel-Cantelli lemma for dynamical systems with exponential decay of correlations

Abstract

Let $([0,1{]}^d,T,\hat{I}¼)$ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $\hat{I}¼$ is absolutely continuous with a density function $h\in L^q({\mathcal{L}}^d)$ for some $q>1$, where ${\mathcal{L}}^d$ is the $d$-dimensional Lebesgue measure. Under mild conditions on the underlying dynamical system, we obtain a strong dynamical Borel-Cantelli lemma for recurrence: For any sequence $\{R_n\}$ of hyperrectangles with sides parallel to the axes and centered at the origin, \[\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}χ_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $\hat{I}¼$-a.e.$\text{x}$},\] where $\text{x}\in [0,1{]}^d$ and $R_k+\text{x}$ is the translation of $R_k$. The result applies to Gauss map, ${\hat{I}}^2$-transformation and expanding toral endomorphisms.