Let $(h_t)_{ t\in \mathbb{ R}}$ be the horocycle flow acting on $(M,\mu)=(\Gamma \backslash \text{ SL}(2,\mathbb{ R}),\mu)$, where $\Gamma$ 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 $$\ \left|\int_M \prod_{ i=0}^{ 2} f_i\circ h^\tau_{ t_i} d \mu^\tau -\prod_{ i=0}^{ 2}\int_M f_i d \mu^\tau \right|\leq C \left(\prod_{ i=0}^{ 2} \|f_i\|_6\right) \left(\min_{ 0\leq i<j\leq 2} |t_i-t_j|\right)^{ -\gamma}.$$ With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $\tau$ being fully supported on the discrete series.
Categories: math.DS