# American Institute of Mathematical Sciences

September  2020, 40(9): 5347-5371. doi: 10.3934/dcds.2020230

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

 1 Department of Mathematics, University of Maryland, 4305 Kirwan Hall, College Park, MD 20742-4015, USA 2 School of Mathematics, Monash University, 9 Rainforest Walk, Clayton, Victoria 3800, Australia

Received  October 2019 Revised  November 2019 Published  June 2020

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 , 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
With the same techniques, we establish polynomial mixing of all orders under the additional assumption of
 $\tau$
being fully supported on the discrete series.
Citation: Adam Kanigowski, Davide Ravotti. Polynomial 3-mixing for smooth time-changes of horocycle flows. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5347-5371. doi: 10.3934/dcds.2020230
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