A NOTE ON BOREL–CANTELLI LEMMAS FOR NON-UNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS

Abstract. Let (Bi) be a sequence of measurable sets in a probability space (X,B, µ) such that ∑∞n=1 µ(Bi) = ∞. The classical Borel–Cantelli lemma states that if the setsBi are independent, then µ({x ∈ X: x ∈ Bi infinitely often (i.o.) ) = 1. Suppose (T,X, µ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether T ix ∈ Bi for µ a.e. x ∈ X and if so, is there an asymptotic estimate on the rate of entry. If T ix ∈ Bi infinitely often for µ a.e. x we call the sequence Bi a Borel–Cantelli sequence. If the sets Bi: = B(p, ri) are nested balls about a point p then the question of whether T ix ∈ Bi infinitely often for µ a.e. x is often called the shrinking target problem. We show, under certain assumptions on the measure µ, that for balls Bi if µ(Bi) ≥ i−γ, 0 < γ < 1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observations implies that the sequence is Borel–Cantelli. If µ(Bi) ≥ C log ii then exponential decay of correlations implies that the sequence is Borel–Cantelli. If it is only assumed that µ(Bi) ≥ 1i then we give conditions in terms of return time statistics which imply that for µ a.e. p sequences of nested balls B(p, 1/i) are Borel–Cantelli. Corollaries of our results are that for planar dispersing billiards and Lozi maps µ a.e. p sequences of nested balls B(p, 1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems. 1