Local mixing on abelian covers of hyperbolic surfaces with cusps

TL;DR

This paper establishes local mixing properties for geodesic flows on abelian covers of hyperbolic surfaces with cusps, extending previous work and applying results to counting problems and prime geodesics.

Contribution

It proves the local mixing theorem for these flows and explores applications to counting and prime geodesic theorems, advancing understanding of hyperbolic surface dynamics.

Findings

Proved local mixing for geodesic flows on abelian covers with cusps

Derived applications to counting problems and prime geodesic theorem

Extended previous work of Oh-Pan on hyperbolic surfaces

Abstract

We prove the local mixing theorem for geodesic flows on abelian covers finite volume hyperbolic surfaces with cusps, which is a continuation of the work of Oh-Pan. We also describe applications to counting problems and the prime geodesic theorem.