Spectral properties of horocycle flows for surfaces of constant negative curvature

TL;DR

This paper proves that certain flows related to horocycle flows on negatively curved surfaces have strong mixing properties and purely absolutely continuous spectra, extending previous results to less regular time changes.

Contribution

It provides a new, concise proof of strong mixing and spectral properties of $W^{ m u}$ flows, including for time changes of horocycle flows with lower regularity.

Findings

$W^{ m u}$ flows are strongly mixing under regularity assumptions.

$W^{ m u}$ flows have purely absolutely continuous spectrum outside constants.

Time changes of horocycle flows with slightly less than $C^2$ regularity have purely absolutely continuous spectrum.

Abstract

We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than $C^2$. This generalises recent results on time changes of horocycle flows.