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

TL;DR

This paper studies Abelian covers of hyperbolic surfaces, proving asymptotic correlation expansions for horocycle flows and demonstrating spectral measure convergence, revealing deep mixing properties and spectral behavior.

Contribution

It introduces new asymptotic expansions for correlations and spectral measures in Abelian covers, advancing understanding of horocycle flow dynamics and spectral theory.

Findings

Proved asymptotic correlation expansion for horocycle flows on Abelian covers.

Established weak convergence of spectral measures to an absolutely continuous measure.

Demonstrated strong mixing properties in the infinite volume setting.

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.