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Abstract. We study the invariant distributions for horocycle maps on Γ\SL(2, R) and prove Sobolev estimates for the cohomological equation of horocycle maps. As an application, we obtain a rate of equidistribution for horocycle maps on compact manifolds.

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Let M = Γ\ PSL(2, R) be a compact manifold, and let f ∈ C ∞ (M ) be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of f along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on M .

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