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

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

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

Adam Kanigowski ^1,, and Davide Ravotti ^2,

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

^* Corresponding author: Adam Kanigowski

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<t_1<t_2 $

, 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<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.

Keywords: Smooth time-changes, horocycle flow, multiple mixing, polynomial mixing, shearing.

Mathematics Subject Classification: 37A10, 37A25, 37C40.

Citation: Adam Kanigowski, Davide Ravotti. Polynomial 3-mixing for smooth time-changes of horocycle flows. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5347-5371. doi: 10.3934/dcds.2020230

References:

[1]	A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, preprint, arXiv: 1905.11628.
[2]	A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of Heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. doi: 10.4310/jdg/1335207373.
[3]	M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, preprint, arXiv: 1701.00945.
[4]	A. Bufetov and G. Forni, Limit theorem for horocycle flows, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851-903. doi: 10.24033/asens.2229.
[5]	D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713. doi: 10.1007/s10955-016-1689-3.
[6]	B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503. doi: 10.24033/bsmf.2405.
[7]	B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows of the torus, submitted.
[8]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.
[9]	L. Flaminio and G. Forni, Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows, Electron. Res. Announc. Math. Sci., 26 (2019), 16-23. doi: 10.3934/era.2019.26.002.
[10]	G. Forni and A. Kanigowski, Time-changes of Heisenberg nilflows, preprint, arXiv: 1711.05543.
[11]	G. Forni and A. Kanigowski, Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows, J. Éc. Polytech. Math., 7 (2020), 63-91. doi: 10.5802/jep.111.
[12]	G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273. doi: 10.3934/jmd.2012.6.251.
[13]	H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics, Lecture Notes in Math., 318, Springer, Berlin, 1973, 95-115.
[14]	B. M. Gurevič, The entropy of horocycle flows, Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.
[15]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.
[16]	B. Host, Mixing of all orders and pairwise independent joinings of system with singular spectrum, Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.
[17]	A. Kanigowski, M. Lemańczyk and C. Ulcigrai, On disjointness properties of some parabolic flows, Invent. Math., (2020). doi: 10.1007/s00222-019-00940-y.
[18]	A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006,649-743. doi: 10.1016/S1874-575X(06)80036-6.
[19]	B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105. doi: 10.2307/1971026.
[20]	B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.
[21]	J. Moreira, The horocycle flow is mixing of all orders, 2015. Available from: https://joelmoreira.wordpress.com/2015/03/08/the-horocycle-flow-is-mixing-of-all-orders/.
[22]	O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspekhi Matem. Nauk (N.S.), 8 (1953), 125-126.
[23]	M. Ratner, Rigidity of horocycle flows, Ann. of Math., 115 (1982), 597-614. doi: 10.2307/2007014.
[24]	M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313. doi: 10.2307/2007030.
[25]	M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32. doi: 10.1007/BF02399199.
[26]	M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288. doi: 10.1017/S0143385700004004.
[27]	M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374. doi: 10.1007/BF01388912.
[28]	D. Ravotti, Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows, Ergodic Theory Dynam. Systems, 39 (2019), 3407-3436. doi: 10.1017/etds.2018.19.
[29]	D. Ravotti, Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces, Annales Henri Poincaré, 18 (2017), 3815-3861. doi: 10.1007/s00023-017-0619-5.
[30]	R. Tiedra de Aldecoa, Spectral analysis of time changes of horocycle flow, J. Mod. Dyn., 6 (2012), 275-285. doi: 10.3934/jmd.2012.6.275.

show all references

References:

[1]	A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, preprint, arXiv: 1905.11628.
[2]	A. Avila, G. Forni and C. Ulcigrai, Mixing for time-changes of Heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. doi: 10.4310/jdg/1335207373.
[3]	M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, preprint, arXiv: 1701.00945.
[4]	A. Bufetov and G. Forni, Limit theorem for horocycle flows, Ann. Sci. Éc. Norm. Supér. (4), 47 (2014), 851-903. doi: 10.24033/asens.2229.
[5]	D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713. doi: 10.1007/s10955-016-1689-3.
[6]	B. Fayad, Polynomial decay of correlations for a class of smooth flows on the two torus, Bull. Soc. Math. France, 129 (2001), 487-503. doi: 10.24033/bsmf.2405.
[7]	B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows of the torus, submitted.
[8]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.
[9]	L. Flaminio and G. Forni, Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows, Electron. Res. Announc. Math. Sci., 26 (2019), 16-23. doi: 10.3934/era.2019.26.002.
[10]	G. Forni and A. Kanigowski, Time-changes of Heisenberg nilflows, preprint, arXiv: 1711.05543.
[11]	G. Forni and A. Kanigowski, Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows, J. Éc. Polytech. Math., 7 (2020), 63-91. doi: 10.5802/jep.111.
[12]	G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273. doi: 10.3934/jmd.2012.6.251.
[13]	H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics, Lecture Notes in Math., 318, Springer, Berlin, 1973, 95-115.
[14]	B. M. Gurevič, The entropy of horocycle flows, Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.
[15]	G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542. doi: 10.1215/S0012-7094-36-00246-6.
[16]	B. Host, Mixing of all orders and pairwise independent joinings of system with singular spectrum, Israel J. Math., 76 (1991), 289-298. doi: 10.1007/BF02773866.
[17]	A. Kanigowski, M. Lemańczyk and C. Ulcigrai, On disjointness properties of some parabolic flows, Invent. Math., (2020). doi: 10.1007/s00222-019-00940-y.
[18]	A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006,649-743. doi: 10.1016/S1874-575X(06)80036-6.
[19]	B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105. doi: 10.2307/1971026.
[20]	B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209. doi: 10.1007/BF01390274.
[21]	J. Moreira, The horocycle flow is mixing of all orders, 2015. Available from: https://joelmoreira.wordpress.com/2015/03/08/the-horocycle-flow-is-mixing-of-all-orders/.
[22]	O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspekhi Matem. Nauk (N.S.), 8 (1953), 125-126.
[23]	M. Ratner, Rigidity of horocycle flows, Ann. of Math., 115 (1982), 597-614. doi: 10.2307/2007014.
[24]	M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313. doi: 10.2307/2007030.
[25]	M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32. doi: 10.1007/BF02399199.
[26]	M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288. doi: 10.1017/S0143385700004004.
[27]	M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374. doi: 10.1007/BF01388912.
[28]	D. Ravotti, Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows, Ergodic Theory Dynam. Systems, 39 (2019), 3407-3436. doi: 10.1017/etds.2018.19.
[29]	D. Ravotti, Quantitative mixing for locally Hamiltonian flows with saddle loops on compact surfaces, Annales Henri Poincaré, 18 (2017), 3815-3861. doi: 10.1007/s00023-017-0619-5.
[30]	R. Tiedra de Aldecoa, Spectral analysis of time changes of horocycle flow, J. Mod. Dyn., 6 (2012), 275-285. doi: 10.3934/jmd.2012.6.275.

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