## Abstract

Let (h_{t})_{t∈ℝ}be the horocycle ow 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 ow (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 have (Equation Presented). With the same techniques, we establish polynomial mixing of all orders under the additional assumption of τ being fully supported on the discrete series.

Original language | English |
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Pages (from-to) | 5347-5371 |

Number of pages | 25 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 40 |

Issue number | 9 |

DOIs | |

Publication status | Published - Sep 2020 |

## Keywords

- Horocycle ow
- Multiple mixing
- Polynomial mixing
- Shearing
- Smooth time-changes