Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows

Abstract

We consider Abelian covers of compact hyperbolic surfaces. We establish an asymptotic expansion of the correlations for the horocycle flow on ${\mathbb{Z}}^d$-covers, thus proving a strong form of Krickeberg mixing. We also prove that the spectral measures around $0$ of the Casimir operators on any increasing sequence of finite Abelian covers converge weakly to an absolutely continuous measure.

27 pages. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it