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 the $\mathbb {Z}$-cover of compact smooth surfaces of negative curvature.

We study the ergodic properties of the translation surface $X_{\unicode{x3bb} ,\mu }$ formed by gluing two flat tori along a slit with holonomy $(\unicode{x3bb} ,\mu ) \in \mathbb {R}^2$. Extending the dichotomy result of Cheung, Hubert, and Masur for the case $\mu = 0$, we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set $\operatorname {\mathrm {NE}}(X_{\unicode{x3bb} ,\mu },\omega )$ of non-ergodic directions is either $0$ or $\frac 12$. This dichotomy is completely characterized by the Pérez–Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the Pérez–Marco condition for best approximation denominators is norm-independent.

In this paper, we prove a quantitative equidistribution theorem for polynomial sequences in a nilmanifold, where the average is taken along spheres instead of cubes. To be more precise, let $\Omega \subseteq \mathbb {Z}^{d}$ be the preimage of a sphere $\mathbb {F}_{p}^{d}$ under the natural embedding from $\mathbb {Z}^{d}$ to $\mathbb {F}_{p}^{d}$. We show that if a rational polynomial sequence $(g(n)\Gamma )_{n\in \Omega }$ is not equidistributed on a nilmanifold $G/\Gamma $, then there exists a non-trivial horizontal character $\eta $ of $G/\Gamma $ such that $\eta \circ g \,\mod \mathbb {Z}$ vanishes on $\Omega $.