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

TL;DR

This paper establishes a strong dynamical Borel-Cantelli lemma for systems with exponential decay of correlations, describing recurrence properties and convergence behavior for hyperrectangles under certain conditions.

Contribution

It introduces a new strong Borel-Cantelli lemma for dynamical systems with exponential decay, extending recurrence results to systems with absolutely continuous measures.

Findings

Proves a strong recurrence property for hyperrectangles in systems with exponential decay.

Shows convergence of normalized recurrence counts to the density function almost everywhere.

Applies results to Gauss map, β-transformation, and expanding toral endomorphisms.

Abstract

Let $ ([0,1]^d,T,\mu) $ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $ \mu $ 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, \[\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\chi_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $ \mu $-a.e.$\textbf{x}$},\] where $ \textbf{x}\in[0,1]^d $ and $ R_k+\textbf{x} $ is the translation of $ R_k $. The result applies to Gauss map, $\beta$-transformation and expanding toral endomorphisms.