  We show that the ergodic averages for the horocycle flow on the two-torus
associated by Giulietti and Liverani to an Anosov diffeomorphism either grow
linearly or are bounded, in other words there are no deviations. For this, we
use topological invariance of the Artin-Mazur zeta function to exclude
resonances outside of the open unit disc. Transfer operators acting on suitable
spaces of anisotropic distributions and their Ruelle determinants are the key
tools in the proof. As a bonus, we show that for any smooth Anosov
diffeomorphism F on the two-torus, the correlations for the measure of maximal
entropy and smooth observables decay with a rate strictly smaller than
exp(-h_top(F)). We compare our results with related work of Forni.
