Quantitative recurrence properties and strong dynamical Borel–Cantelli lemmas for dynamical systems with exponential decay of correlations

Abstract Let $ ([0,1]^d,T,\mu ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder 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 suitable conditions on the underlying dynamical system, we obtain a strong dynamical Borel–Cantelli lemma for recurrence: for any sequence $ \{R_n\} $ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to $0$ as $n\to \infty $ , where $ \mathbf {x}\in [0,1]^d $ and $ R_n+\mathbf {x} $ is the translation of $ R_n $ . The result applies to the Gauss map, $\beta $ -transformations, and expanding toral endomorphisms.