The Cohomological Equation and Invariant Distributions for Horocycle Maps

TL;DR

This paper investigates invariant distributions and Sobolev estimates for the cohomological equation of horocycle maps on certain manifolds, providing insights into their equidistribution rates.

Contribution

It introduces new Sobolev estimates for the cohomological equation of horocycle maps and applies these to quantify equidistribution rates on compact manifolds.

Findings

Established Sobolev estimates for the cohomological equation

Derived rate of equidistribution for horocycle maps

Analyzed invariant distributions for horocycle maps

Abstract

We study the invariant distributions for the horocycle map on $\Gamma\backslash SL(2, \mathbb{R})$ and prove Sobolev estimates for the cohomological equation of the horocycle map. As an application, we obtain an estimate for the rate of equidistribution for horocycle maps on compact manifolds.