-
Previous Article
What is topological about topological dynamics?
- DCDS Home
- This Issue
-
Next Article
Non-formally integrable centers admitting an algebraic inverse integrating factor
Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$
USA |
We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $ Γ \backslash{\rm{PSL}}(2, \mathbb{R})^d$, where $ d ∈ \mathbb{N}_{≥2}$ and $ Γ \subset {\rm{PSL}}(2, \mathbb{R})^d$ is a cocompact and irreducible lattice. As a consequence, we prove exponential multiple mixing for partially hyperbolic coordinate geodesic flows on these manifolds.
References:
[1] |
M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative Multiple Mixing, to appear in J. Eur. Math. Soc. (JEMS) |
[2] |
M. Brin and Y. Pessin,
Flows of frames on manifolds of negative curvature, Uspehi Mat. Nauk., 28 (1973), 209-210.
|
[3] |
M. Brin and Y. Pessin,
Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
|
[4] |
T. Browning and Ilya Vinogradov,
Effective Ratner theorem for $ {\rm{ASL}}(2, \mathbb{R})$ and gaps in $ \sqrt{n}$ modulo 1, J. London Math. Soc., 94 (2016), 61-84.
|
[5] |
S. G. Dani,
Kolmogorov automorphisms on homogeneous spaces, Amer. J. Math., 98 (1976), 119-163.
doi: 10.2307/2373618. |
[6] |
S. G. Dani,
Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155.
doi: 10.1215/S0012-7094-77-04407-6. |
[7] |
D. Dolgopyat,
Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society, 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
[8] |
D. Dolgopyat,
On Decay of correlations in Anosov flows, Annals of Math., 147 (1998), 357-390.
doi: 10.2307/121012. |
[9] |
L. Flaminio and G. Forni,
Invariant Distributions and Time Averages for Horocycle Flows, Duke J. of Math., 119 (2003), 465-526.
doi: 10.1215/S0012-7094-03-11932-8. |
[10] |
L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv:1407.3640 |
[11] |
L. Flaminio, G. Forni and J. Tanis,
Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (5), 1359-1448.
|
[12] |
A. Gorodnik and R. Spatzier,
Exponential mixing of nilmanifold automorphsims, Journal d'Analyse Methematique, 123 (2014), 355-396.
doi: 10.1007/s11854-014-0024-7. |
[13] |
D. Kelmer and P. Sarnak,
Strong spectral gaps for compact quotients of products of $ {\rm{PSL}}(2, \mathbb R)$, J. Eur. Math. Soc., 11 (2009), 283-313.
|
[14] |
I. Konstantoulas,
Effective decay of multiple correlations in semidirect product actions, Journal of Modern Dynamics, 10 (2016), 81-111.
doi: 10.3934/jmd.2016.10.81. |
[15] |
C. Liverani,
On Contact Anosov flows, Annals of Math., 159 (2004), 1275-1312.
doi: 10.4007/annals.2004.159.1275. |
[16] |
E. Nelson,
Analytic vectors, Annals of Math., 70 (1959), 572-615.
doi: 10.2307/1970331. |
[17] |
A. Strombergsson,
An Effective Ratner Equidistribution Result for $ {\rm{SL}}(2,\mathbb R)\ltimes \mathbb R^2$, Duke Math. J., 164 (2015), 843-902.
doi: 10.1215/00127094-2885873. |
[18] |
J. Tanis and P. Vishe,
Uniform bounds for period integrals and sparse equidistribution, International Mathematics Research Notices, (2015), 13728-13756.
doi: 10.1093/imrn/rnv115. |
[19] |
J. Tanis, Effective equidistribution for some unipotent flows in $ {\rm{PSL}}(2, \mathbb{R})^k$ mod cocompact irreducible lattice, arXiv:1412.5353v3 |
[20] |
A. Venkatesh,
Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., 172 (2010), 989-1094.
doi: 10.4007/annals.2010.172.989. |
[21] |
I. Ilya Vinogradov, Effective equidistribution of horocycle lifts, arXiv:1607.04769 |
show all references
References:
[1] |
M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative Multiple Mixing, to appear in J. Eur. Math. Soc. (JEMS) |
[2] |
M. Brin and Y. Pessin,
Flows of frames on manifolds of negative curvature, Uspehi Mat. Nauk., 28 (1973), 209-210.
|
[3] |
M. Brin and Y. Pessin,
Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
|
[4] |
T. Browning and Ilya Vinogradov,
Effective Ratner theorem for $ {\rm{ASL}}(2, \mathbb{R})$ and gaps in $ \sqrt{n}$ modulo 1, J. London Math. Soc., 94 (2016), 61-84.
|
[5] |
S. G. Dani,
Kolmogorov automorphisms on homogeneous spaces, Amer. J. Math., 98 (1976), 119-163.
doi: 10.2307/2373618. |
[6] |
S. G. Dani,
Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155.
doi: 10.1215/S0012-7094-77-04407-6. |
[7] |
D. Dolgopyat,
Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society, 356 (2004), 1637-1689.
doi: 10.1090/S0002-9947-03-03335-X. |
[8] |
D. Dolgopyat,
On Decay of correlations in Anosov flows, Annals of Math., 147 (1998), 357-390.
doi: 10.2307/121012. |
[9] |
L. Flaminio and G. Forni,
Invariant Distributions and Time Averages for Horocycle Flows, Duke J. of Math., 119 (2003), 465-526.
doi: 10.1215/S0012-7094-03-11932-8. |
[10] |
L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv:1407.3640 |
[11] |
L. Flaminio, G. Forni and J. Tanis,
Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (5), 1359-1448.
|
[12] |
A. Gorodnik and R. Spatzier,
Exponential mixing of nilmanifold automorphsims, Journal d'Analyse Methematique, 123 (2014), 355-396.
doi: 10.1007/s11854-014-0024-7. |
[13] |
D. Kelmer and P. Sarnak,
Strong spectral gaps for compact quotients of products of $ {\rm{PSL}}(2, \mathbb R)$, J. Eur. Math. Soc., 11 (2009), 283-313.
|
[14] |
I. Konstantoulas,
Effective decay of multiple correlations in semidirect product actions, Journal of Modern Dynamics, 10 (2016), 81-111.
doi: 10.3934/jmd.2016.10.81. |
[15] |
C. Liverani,
On Contact Anosov flows, Annals of Math., 159 (2004), 1275-1312.
doi: 10.4007/annals.2004.159.1275. |
[16] |
E. Nelson,
Analytic vectors, Annals of Math., 70 (1959), 572-615.
doi: 10.2307/1970331. |
[17] |
A. Strombergsson,
An Effective Ratner Equidistribution Result for $ {\rm{SL}}(2,\mathbb R)\ltimes \mathbb R^2$, Duke Math. J., 164 (2015), 843-902.
doi: 10.1215/00127094-2885873. |
[18] |
J. Tanis and P. Vishe,
Uniform bounds for period integrals and sparse equidistribution, International Mathematics Research Notices, (2015), 13728-13756.
doi: 10.1093/imrn/rnv115. |
[19] |
J. Tanis, Effective equidistribution for some unipotent flows in $ {\rm{PSL}}(2, \mathbb{R})^k$ mod cocompact irreducible lattice, arXiv:1412.5353v3 |
[20] |
A. Venkatesh,
Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., 172 (2010), 989-1094.
doi: 10.4007/annals.2010.172.989. |
[21] |
I. Ilya Vinogradov, Effective equidistribution of horocycle lifts, arXiv:1607.04769 |
[1] |
Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42. |
[2] |
Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90 |
[3] |
Raf Cluckers, Julia Gordon, Immanuel Halupczok. Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups. Electronic Research Announcements, 2014, 21: 137-152. doi: 10.3934/era.2014.21.137 |
[4] |
Palle Jorgensen, James Tian. Harmonic analysis of network systems via kernels and their boundary realizations. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021105 |
[5] |
Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757 |
[6] |
Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121 |
[7] |
Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks and Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295 |
[8] |
Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271 |
[9] |
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 |
[10] |
Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953 |
[11] |
C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems and Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457 |
[12] |
John R. Tucker. Attractors and kernels: Linking nonlinear PDE semigroups to harmonic analysis state-space decomposition. Conference Publications, 2001, 2001 (Special) : 366-370. doi: 10.3934/proc.2001.2001.366 |
[13] |
Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201 |
[14] |
Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291 |
[15] |
Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure and Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425 |
[16] |
Tomás Caraballo, Maria-José Garrido-Atienza, Javier López-de-la-Cruz, Alain Rapaport. Modeling and analysis of random and stochastic input flows in the chemostat model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3591-3614. doi: 10.3934/dcdsb.2018280 |
[17] |
Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137 |
[18] |
Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations and Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193 |
[19] |
Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551 |
[20] |
Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]