The Cohomological Equation and Invariant Distributions for Horocycle   Maps
  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. 
James Tanis
math.DS