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We show that the ergodic averages for the horocycle flow on the two-torusassociated by Giulietti and Liverani to an Anosov diffeomorphism either growlinearly or are bounded, in other words there are no deviations. For this, weuse topological invariance of the Artin-Mazur zeta function to excluderesonances outside of the open unit disc. Transfer operators acting on suitablespaces of anisotropic distributions and their Ruelle determinants are the keytools in the proof. As a bonus, we show that for any smooth Anosovdiffeomorphism F on the two-torus, the correlations for the measure of maximalentropy and smooth observables decay with a rate strictly smaller thanexp(-h_top(F)). We compare our results with related work of Forni.

There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus