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

Abstract

Let \begin{document}$ (h_t)_{t\in {\mathbb{R}}} $\end{document} be the horocycle flow acting on \begin{document}$ (M,\mu) = (\Gamma \backslash \operatorname{SL}(2,{\mathbb{R}}), \mu) $\end{document} , where \begin{document}$ \Gamma $\end{document} is a co-compact lattice in \begin{document}$ \operatorname{SL}(2,{\mathbb{R}}) $\end{document} and \begin{document}$ \mu $\end{document} is the homogeneous probability measure locally given by the Haar measure on \begin{document}$ \operatorname{SL}(2,{\mathbb{R}}) $\end{document} . Let \begin{document}$ \tau\in W^6(M) $\end{document} be a strictly positive function and let \begin{document}$ \mu^{\tau} $\end{document} be the measure equivalent to \begin{document}$ \mu $\end{document} with density \begin{document}$ \tau $\end{document} . We consider the time changed flow \begin{document}$ (h_t^\tau)_{t\in {\mathbb{R}}} $\end{document} and we show that there exists \begin{document}$ \gamma = \gamma(M,\tau)>0 $\end{document} and a constant \begin{document}$ C>0 $\end{document} such that for any \begin{document}$ f_0, f_1, f_2\in W^6(M) $\end{document} and for all \begin{document}$ 0 = t_0 , we have \begin{document}$ \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 With the same techniques, we establish polynomial mixing of all orders under the additional assumption of \begin{document}$ \tau $\end{document} being fully supported on the discrete series.