Polynomial 3-mixing for smooth time-changes of horocycle flows

Let $ (h_t)_{t\in {\mathbb{R}}} $ be the horocycle flow acting on $ (M,\mu) = (\Gamma \backslash \operatorname{SL}(2,{\mathbb{R}}), \mu) $, where $ \Gamma $ is a co-compact lattice in $ \operatorname{SL}(2,{\mathbb{R}}) $ and $ \mu $ is the homogeneous probability measure locally given by the Haar measure on $ \operatorname{SL}(2,{\mathbb{R}}) $. Let $ \tau\in W^6(M) $ be a strictly positive function and let $ \mu^{\tau} $ be the measure equivalent to $ \mu $ with density $ \tau $. We consider the time changed flow $ (h_t^\tau)_{t\in {\mathbb{R}}} $ and we show that there exists $ \gamma = \gamma(M,\tau)>0 $ and a constant $ C>0 $ such that for any $ f_0, f_1, f_2\in W^6(M) $ and for all $ 0 = t_0<t_1<t_2 $, we have

$ \left|\int_M \prod\limits_{i = 0}^{2} f_i\circ h^\tau_{t_i} {\rm{d}} \mu^\tau -\prod\limits_{i = 0}^{2}\int_M f_i {\rm{d}} \mu^\tau \right|\leq C \left(\prod\limits_{i = 0}^{2} \|f_i\|_6\right) \left(\min\limits_{0\leq i<j\leq 2} |t_i-t_j|\right)^{-\gamma}. $

With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $ \tau $ being fully supported on the discrete series.