Quantitative recurrence for Z-extension of three-dimensional Axiom A flows

TL;DR

This paper investigates the recurrence behavior of Z-extensions of Axiom A flows, analyzing return times and their distribution, with applications to geodesic flows on negatively curved surfaces.

Contribution

It provides new results on the asymptotic distribution of return times for Z-extensions of Axiom A flows, including convergence properties.

Findings

Almost everywhere convergence of return times

Convergence in distribution for measures absolutely continuous w.r.t. the invariant measure

Applicability to geodesic flows on negatively curved surfaces

Abstract

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.