-
Previous Article
Liouville theorems on the upper half space
- DCDS Home
- This Issue
-
Next Article
Reducibility of quasi-periodically forced circle flows
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 |
$ (h_t)_{t\in {\mathbb{R}}} $ |
$ (M,\mu) = (\Gamma \backslash \operatorname{SL}(2,{\mathbb{R}}), \mu) $ |
$ \Gamma $ |
$ \operatorname{SL}(2,{\mathbb{R}}) $ |
$ \mu $ |
$ \operatorname{SL}(2,{\mathbb{R}}) $ |
$ \tau\in W^6(M) $ |
$ \mu^{\tau} $ |
$ \mu $ |
$ \tau $ |
$ (h_t^\tau)_{t\in {\mathbb{R}}} $ |
$ \gamma = \gamma(M,\tau)>0 $ |
$ C>0 $ |
$ f_0, f_1, f_2\in W^6(M) $ |
$ 0 = t_0<t_1<t_2 $ |
$ \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}. $ |
$ \tau $ |
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. |
[1] |
Giovanni Forni, Corinna Ulcigrai. Time-changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 251-273. doi: 10.3934/jmd.2012.6.251 |
[2] |
Livio Flaminio, Giovanni Forni. Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows. Electronic Research Announcements, 2019, 26: 16-23. doi: 10.3934/era.2019.26.002 |
[3] |
Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275 |
[4] |
Siyuan Tang. New time-changes of unipotent flows on quotients of Lorentz groups. Journal of Modern Dynamics, 2022, 18: 13-67. doi: 10.3934/jmd.2022002 |
[5] |
Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012 |
[6] |
Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13 |
[7] |
Arnaud Goullet, Ian Glasgow, Nadine Aubry. Dynamics of microfluidic mixing using time pulsing. Conference Publications, 2005, 2005 (Special) : 327-336. doi: 10.3934/proc.2005.2005.327 |
[8] |
Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61 |
[9] |
Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315 |
[10] |
Kazuhiko Yamamoto, Kiyoshi Hosono, Hiroko Nakayama, Akio Ito, Yuichi Yanagi. Experimental data for solid tumor cells: Proliferation curves and time-changes of heat shock proteins. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 235-244. doi: 10.3934/dcdss.2012.5.235 |
[11] |
Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001 |
[12] |
Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082 |
[13] |
A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195 |
[14] |
Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259 |
[15] |
James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042 |
[16] |
Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 |
[17] |
Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63. |
[18] |
Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181 |
[19] |
Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 |
[20] |
Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]