Strong Borel–Cantelli lemmas for recurrence
Abstract
Let $(X,T,\mu ,d)$ 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 $\mu (B_k(x)) = M_k$. Under a short return time assumption, we prove a strong Borel–Cantelli lemma, including an error term, for recurrence, i.e., for $\mu $-a.e. $x \in X$, $$\sum _{k=1}^{n}\ \mathbf 1_{B_k(x)} (T^k x) = \varPhi (n) + O\big( \varPhi (n)^{1/2} (\log \varPhi (n))^{3/2 + \varepsilon} \big), $$ where $\varPhi (n) = \sum _{k=1}^{n} \mu (B_k(x))$. Applications to systems include some non-linear piecewise expanding interval maps and hyperbolic automorphisms of $\mathbb T^2$.
