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

Abstract

Let (ht) t∈R be the horocycle flow acting on (M, µ) = (Γ\ SL(2, R), µ), where Γ is a co-compact lattice in SL(2, R) and µ is the homogeneous probability measure locally given by the Haar measure on SL(2, R). Let τ ∈ 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 τ t ) t∈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 ∈ W 6 (M ) and for all 0 = t 0 < t 1 < t 2 , we haveWith the same techniques, we establish polynomial mixing of all orders under the additional assumption of τ being fully supported on the discrete series.