Strong Borel--Cantelli Lemmas for Recurrence

Abstract

Let $(X,T,\hat{I}¼,d)$ be a metric measure-preserving system for which $3$-fold correlations decay exponentially for Lipschitz continuous observables. Suppose that $(M_k)$ is a sequence satisfying some weak decay conditions and suppose there exist open balls $B_k(x)$ around $x$ such that $\hat{I}¼(B_k(x))=M_k$. Under a short return time assumption, we prove a strong Borel--Cantelli lemma, including an error term, for recurrence, i.e., for $\hat{I}¼$-a.e. $x\in X$, \[ \sum_{k=1}^{n} \mathbf{1}_{B_k(x)} (T^k x) = Φ(n) + O \bigl( Φ(n)^{1/2} (\log Φ(n))^{3/2 + \varepsilon} \bigr), \] where $\hat{I}¦(n)={\sum }_{k=1}^n\hat{I}¼(B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of ${\mathbf{T}}^2$.

22 pages, 3 figures