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

Let ([0,1]ᵈ,T,μ ) be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that μ is absolutely continuous with a density function h∈ Lq( Lᵈ) for some $ q>1 $ , where Lᵈ 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 ₙ\ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to $0$ as n→ ∞ , where x∈ [0,1]ᵈ and Rₙ+x is the translation of Rₙ . The result applies to the Gauss map, β -transformations, and expanding toral endomorphisms.