Spectral properties of horocycle flows for surfaces of negative curvature

Rafael Tiedra de Aldecoa

Published 2015-06-21Version 1

We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity asumptions, 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 a particular case, we obtain that a class of horocycle flows for compact surfaces of (possibly variable) negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This generalises recent results on time changes of the classical horocycle flows for compact surfaces of constant negative curvature.