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

Let $(h_t{)}_{t\in \mathbb{R}}$ be the horocycle flow acting on $(M,\mu )=(\mathrm{\Gamma }\mathrm{\backslash }\text{SL}(2,\mathbb{R}),\mu )$, where $\mathrm{\Gamma }$ is a co-compact lattice in $\text{SL}(2,\mathbb{R})$ and $\mu$ is the homogeneous probability measure locally given by the Haar measure on $\text{SL}(2,\mathbb{R})$. 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 $$\mid {\int }_M\prod _{i=0}^2{f}_i\circ h_{t_i}^{\tau }d{\mu }^{\tau }-\prod _{i=0}^2{\int }_M{f}_{i}d{\mu }^{\tau }\mid \le C\left(\prod _{i=0}^2\mathrm{\parallel }{f}_i{\mathrm{\parallel }}_6\right){(\underset{0\le i<j\le 2}{\min }\mathrm{\mid }{t}_i-t_j\mathrm{\mid })}^{-\gamma }\mathrm{.}$$ With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $\tau$ being fully supported on the discrete series.