Asymptotics and limit theorems for horocycle ergodic integrals \`a la Ratner

We apply a method inspired by Ratner's work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987) and developed by Burger (Duke Math. J., 1990) to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni (Duke Math. J., 2003), and we show that the coefficients in the asymptotic expansion are H\"{o}lder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni (Ann. Sci. \'Ec. Norm. Sup\'er., 2014) and, in an appendix by Emilio Corso, of a temporal limit theorem by Dolgopyat and Sarig (J. Stat. Phys., 2017).