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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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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.  Google Scholar

[3]

M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, preprint, arXiv: 1701.00945. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows of the torus, submitted. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Forni and A. Kanigowski, Time-changes of Heisenberg nilflows, preprint, arXiv: 1711.05543. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. M. Gurevič, The entropy of horocycle flows, Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.   Google Scholar

[15]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542.  doi: 10.1215/S0012-7094-36-00246-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026.  Google Scholar

[20]

B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209.  doi: 10.1007/BF01390274.  Google Scholar

[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/. Google Scholar

[22]

O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspekhi Matem. Nauk (N.S.), 8 (1953), 125-126.   Google Scholar

[23]

M. Ratner, Rigidity of horocycle flows, Ann. of Math., 115 (1982), 597-614.  doi: 10.2307/2007014.  Google Scholar

[24]

M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313.  doi: 10.2307/2007030.  Google Scholar

[25]

M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32.  doi: 10.1007/BF02399199.  Google Scholar

[26]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[27]

M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[3]

M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative multiple mixing, preprint, arXiv: 1701.00945. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

B. Fayad, G. Forni and A. Kanigowski, Lebesgue spectrum for area preserving flows of the torus, submitted. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

G. Forni and A. Kanigowski, Time-changes of Heisenberg nilflows, preprint, arXiv: 1711.05543. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

B. M. Gurevič, The entropy of horocycle flows, Dokl. Akad. Nauk SSSR, 136 (1961), 768-770.   Google Scholar

[15]

G. A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J., 2 (1936), 530-542.  doi: 10.1215/S0012-7094-36-00246-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math., 105 (1977), 81-105.  doi: 10.2307/1971026.  Google Scholar

[20]

B. Marcus, The horocycle flow is mixing of all degrees, Invent. Math., 46 (1978), 201-209.  doi: 10.1007/BF01390274.  Google Scholar

[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/. Google Scholar

[22]

O. S. Parasyuk, Flows of horocycles on surfaces of constant negative curvature, Uspekhi Matem. Nauk (N.S.), 8 (1953), 125-126.   Google Scholar

[23]

M. Ratner, Rigidity of horocycle flows, Ann. of Math., 115 (1982), 597-614.  doi: 10.2307/2007014.  Google Scholar

[24]

M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math., 118 (1983), 277-313.  doi: 10.2307/2007030.  Google Scholar

[25]

M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32.  doi: 10.1007/BF02399199.  Google Scholar

[26]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[27]

M. Ratner, Rigid reparametrizations and cohomology for horocycle flows, Invent. Math., 88 (1987), 341-374.  doi: 10.1007/BF01388912.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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