On strong forms of the Borel–Cantelli lemma and intermittent interval maps

We derive new variants of the quantitative Borel–Cantelli lemma and apply them to analysis of statistical properties for some dynamical systems. We consider intermittent maps of (0,1] which have absolutely continuous invariant probability measures. In particular, we prove that every sequence of intervals with left endpoints uniformly separated from zero is the strong Borel–Cantelli sequence with respect to such map and invariant measure.