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A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 18 September 2024, pp. 1-20
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Let
$(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence 
$(M_k)$ that converges to 
$0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for 
$\mu $-almost every 
$x\in X$, where
$$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$
$\mu (B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.
