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

Let $(h_t)_{t\in \mathbb{R}}$ be the horocycle flow acting on $(M,μ)=(Γ\backslash \text{SL}(2,\mathbb{R}),μ)$, where $Γ$ is a co-compact lattice in $\text{SL}(2,\mathbb{R})$ and $μ$ is the homogeneous probability measure locally given by the Haar measure on $\text{SL}(2,\mathbb{R})$. Let $τ\in W^6(M)$ be a strictly positive function and let $μ^τ$ be the measure equivalent to $μ$ with density $τ$. We consider the time changed flow $(h_t^τ)_{t\in \mathbb{R}}$ and we show that there exists $γ=γ(M,τ)>0$ 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 $\ \left|\int_M \prod_{i=0}^{2} f_i\circ h^τ_{t_i} d μ^τ-\prod_{i=0}^{2}\int_M f_i d μ^τ\right|\leq C \left(\prod_{i=0}^{2} \|f_i\|_6\right) \left(\min_{0\leq i<j\leq 2} |t_i-t_j|\right)^{-γ}.$ With the same techniques, we establish polynomial mixing of all orders under the additional assumption of $τ$ being fully supported on the discrete series.

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