2011/03/10 by N. Haydn, M. Nicol, Haydn, N. +5
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.1103.2113
20 pages
arxiv created 2011/03/10 · arxiv updated 2011/03/11
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 sets Bi 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 Ti x∈ Bi for μ a.e. x∈ X and if so, is there an asymptotic estimate on the rate of entry. If Ti x∈ 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 Ti x∈ 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)≥ (Clog i)/(i) then exponential decay of correlations implies that the sequence is Borel-Cantelli. If it is only assumed that μ(Bi) ≥ (1)/(i) 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.