Time-Changes of Horocycle Flows

TL;DR

This paper investigates smooth time-changes of horocycle flows on hyperbolic surfaces, establishing precise bounds on their equidistribution and mixing rates, and analyzing their spectral properties.

Contribution

It provides sharp bounds on equidistribution and mixing rates for smooth time-changed horocycle flows, and characterizes their spectral type as absolutely continuous.

Findings

Sharp bounds on equidistribution rates

Sharp bounds on mixing rates

Spectral analysis shows absolute continuity

Abstract

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.