  In this paper, we study the quantitative recurrence properties in the case of
$\mathbb{Z}$-extension of Axiom A flows on a Riemannian manifold. We study the
asymptotic behavior of the first return time to a small neighborhood of the
starting point. We establish results of almost everywhere convergence, and of
convergence in distribution with respect to any probability measure absolutely
continuous with respect to the infinite invariant measure. In particular, our
results apply to geodesic flows on $\mathbb{Z}$-cover of compact smooth
surfaces of negative curvature.
